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Border Probable Primes around
'Powers of Ten'
rood Border PRP's table (00000-04999) rood Border PRP's table (05000-09999)
rood Border PRP's table (10000-14999) rood Border PRP's table (15000-19999)
rood Border PRP's table (20000-24999) rood Border PRP's table (25000-34999)
rood Border PRP's table (35000-99999)
rood comments


Border PRP's around 'Powers of Ten'
                    10^(x-1)                                                10^x
      x-1             B                          x                            B             x+1
 ————————————————————>O<—————————————————————————————————————————————————————>O<————————————————————
                      R                                                       R
 ---------------o-----D-----o-------------------------------------------o-----D-----o---------------
                      E                                                       E     
               -y     R    +y                                          -y     R    +y

Project Research

Probable primes of the form 10^x ± y.
(y = displacement from axis)

Borderprimes. Closest primes around axes 10^x.

deficient prp : displacement < exponent
efficient prp : displacement = exponent
sufficient prp : displacement > exponent

displacement +y < -y versus +y > -y (statistics)

Search for equal + and - values (equidistant).

PrimeForm/GW (PFGW) Program
[ Extracted from PFGW documentation A.1.]

PrimeForm/GW (PFGW) is a primality tester.
It tests whether or not a number is prime, or at least it tries to.
It is able to test any arbitrary number, though some faster then others.
About the name: PrimeForm speaks for itself, the GW comes from George
Woltman, the person who made the extremely fast modular code available
for this project (well he made it for Prime95, but allows us to use it ;)
It is a command line (DOS Box) utility. It is an 'Open Source' project,
and is ported to Win32, Unix, Linux, BeOS, and possibly other platforms.
The source code can be obtained at [after an initial subscription]:
 http://groups.yahoo.com/group/openpfgw/files/
The most current development version, and the release version of PFGW
for Win32 can be obtained at:
 http://groups.yahoo.com/group/primeform/files/
There are two mailing-lists concerned with PFGW:
 http://groups.yahoo.com/group/primeform/
  which is the "user's group"
and
 http://groups.yahoo.com/group/openpfgw/
  which is the "developer's group".

Motivation

Patrick De Geest (that's me) started to maintain this list already back
in 2004 when confronted with the submissions by Milton L. Brown
in H. Lifchitz's PRP Records table. I felt I needed a more systematic
approach, hence the creation of this collection. Recently Phil Carmody
asked the same after he noted MLB's nice pair 10^12990 + 11163 and
10^12991 + 11163 in Yahoo's PrimeNumbers e-mail discussion list,
read message #18229.

At the same time this is a nice application for PFGW a PrimeForm project
where lots of people can participate and contribute for a long time to come
dependent on how many free cpu cycles one has. Reserve a small or large
range and get the job done. Nevertheless one can join in for just one
borderprime as well. Future generations of primehunters might use this list
of prp's e.g. as a huge store of candidates to test for genuine prime.

Data material

The main table is in text format and the files to be consulted are called
borderprp(00000-04999).txt
borderprp(05000-09999).txt
borderprp(10000-14999).txt
borderprp(15000-19999).txt
borderprp(20000-24999).txt
borderprp(25000-34999).txt
borderprp(35000-99999).txt

Warning when using PFGW !

In February 2017 I thought I had found the following record borderprp and gap:
10^21834 –2928413 and 10^21834 + 2819281
I used pfgw version PFGW Version 3.7.10.64BIT.20150809.Win_Dev [GWNUM 28.6]
Jens Kruse Andersen quickly put me back on my feet by telling me
that the test is false because using any -a1, -a2, -a3, etc. switch on some
smaller displacements like e.g. 10^21834 – 93 results in different residues.
He wrote : "Your pfgw version may have a problem like choosing a bad FFT size
which often gives false results for this size and form. "
Using any -a switch must produce the same residues in order to be valid.
So somewhere during the search a composite was reported that could have
been a 3-PRP! thereby missing the smallest or largest borderprp.
{ In this particular case 10^21834 + 10017 showed up as 3-PRP! by JKA
by running the searchloop with the -a1 switch and his older pfgw version.
confirmed by other methods as well like Pari/GP.}
By the way some time later on I indeed found the other borderprp
10^21834 – 69809 having the smallest displacement.

C:\pfgw box 9>pfgw64 -q"10^21834-93"
10^21834-93 is composite: RES64: [8EF5236AE00E8EE4] (1.4598s+0.0003s)

C:\pfgw box 9>pfgw64 -a1 -q"10^21834-93"
10^21834-93 is composite: RES64: [0C41339B9D1E7C77] (1.9006s+0.0003s)

C:\pfgw box 9>pfgw64 -a2 -q"10^21834-93"
10^21834-93 is composite: RES64: [B92D78FA1D2C9E3A] (2.4376s+0.0003s)

As one can see the RES64 residue values are different from each other.
Again, a composite claim from this test means nothing in this case.



Let me now redo the test on the following borderprp 10^114575 – 581081.
Here PFGW also tells me this is 3-PRP! having used any -a switch
and with all the smaller displacements reported composite.
I choose one of the smaller displacements like for instance 500001
and start the following tests with other -a switches

C:\pfgw box 9>pfgw64 -q"10^114575 – 500001"
10^114575-500001 is composite: RES64: [210AB3A805D8CC8B] (53.3825s+0.0015s)

C:\pfgw box 9>pfgw64 -a1 -q"10^114575-500001"
10^114575-500001 is composite: RES64: [210AB3A805D8CC8B] (63.5933s+0.0026s)

C:\pfgw box 9>pfgw64 -a2 -q"10^114575-500001"
10^114575-500001 is composite: RES64: [210AB3A805D8CC8B] (66.4350s+0.0026s)

Here, the RES64 residue values are all equal or identical.
So here the composite claims smaller than 581081 are not flawed and the 3-PRP!
result for 10^114575-581081 seems to be valid.



From various documentation of pfgw
-a<number> Authenticates composites. Using a -a0 will PRP test using the
"normal" FFT sizes. A switch of -a1 will test using 1 less bit per
FFT limb. This will take more time but will also eliminate any chance of
rounding errors inherant with FFT multiplies. The lower 62 bits of the
results of a PRP test will be listed (on composites only). This allows
checking for math library errors. If a run on the same numbers using -a0
and a -a1 results in different residues, then there were round off errors
.
FFT = Fast Fourier Transform.

Applied a temporary patch to gwnum to address an issue with special modular
reduction on PRP tests. For larger bases gwnum sometimes has an issue with
carry propagation which masks round off errors. For these PRP tests, PFGW
indicates that the number is composite when it is really PRP
. The patched
version of gwnum will detect the problem and choose a larger FFT size before
the PRP test is started. It is recommended that if you are concerned about
the results from older versions of PFGW, that you use the -F option of this
release and compare the output to that of the 3.3.5 release. If you have an
input file, use the -l option to output the chosen FFT size to pfgw.log and
then compare the two files. Numbers with different FFT sizes should be
re-tested.

Added -F option to show the chosen FFT size, but to not perform a test.
This option only applies to PRP tests.



Here is my collection of borderprp's that went haywire using
PFGW Version 3.7.10.64BIT.20150809.Win_Dev [GWNUM 28.6].
These are the genuine probable primes that PFGW reported composite !
If you can add more probable primes to this starting list I would
appreciate it that you notify me.

10^21834– 69809+ 10017
10^23339– 60759+ 59011
10^27102– 2463+ 112863





Sloane's OEIS sequences

A033873, A033874 and A038804
http://oeis.org/A033873
http://oeis.org/A033874
http://oeis.org/A038804
A003617 and A003618
http://oeis.org/A003617
http://oeis.org/A003618
A096548
http://oeis.org/A096548

Initials and abbreviations

AG= Alexander Gramolin
AN= Anand Nair
DA= Dirk Augustin
DH= Daniel Heuer
ET= Edward Trice
HL= Henri Lifchitz
HR= Harald Ritter
JE= Jason Earls
JKA= Jens Kruse Andersen
KEP= Kenneth Egtved Pedersen
LRP= Lelio R. Paula
MHW= Meng-Hsuan Wu
MLB= Milton L. Brown
MO= Mike Oakes
MT= Markus Tervooren
PDG= Patrick De Geest
PK= Peter Kaiser
RC= Ray Chandler
RP= Robert Price
SB= Serge Batalov
TA= Torbjörn Alm
TM= Thomas Masser

ps. borderprp < 10000 will not be copyrighted nor dated
as they are quite easy to reproduce.

tbv = to be verified
vb = verified by

Source material

10^n - 1 is always divisible by 3
http://oeis.org/A089675 Numbers n such that 10^n - 3 is a prime. [>407197]
10^n - 7 is always divisible by 3
http://oeis.org/A095714 Numbers n such that 10^n - 9 is a prime. [>221631]
http://oeis.org/A092767 Numbers n such that 10^n - 11 is a prime. [>100000]
10^n - 13 is always divisible by 3
http://oeis.org/A108326 Numbers n such that 10^n - 17 is a prime. [>100000]
10^n - 19 is always divisible by 3
http://oeis.org/A108327 Numbers n such that 10^n - 21 is a prime. [>100000]
http://oeis.org/A108328 Numbers n such that 10^n - 23 is a prime. [>100000]
http://oeis.org/A108329 Numbers n such that 10^n - 27 is a prime. [>100000]
http://oeis.org/A108330 Numbers n such that 10^n - 29 is a prime. [>100000]
10^n - 31 is always divisible by 3
http://oeis.org/A108364 Numbers n such that 10^n - 33 is a prime. [>100000]
10^n - 37 is always divisible by 3
http://oeis.org/A108365 Numbers n such that 10^n - 39 is a prime. [>100000]
http://oeis.org/A178406 Numbers n such that 10^n - 41 is a prime. [>100000]
10^n - 43 is always divisible by 3
http://oeis.org/A178175 Numbers n such that 10^n - 47 is a prime. [>100000]
10^n - 49 is always divisible by 3
http://oeis.org/A178429 Numbers n such that 10^n - 51 is a prime. [>40000]
http://oeis.org/A178430 Numbers n such that 10^n - 53 is a prime. [>100000]
http://oeis.org/A108493 Numbers n such that 10^n - 57 is a prime. [>100000]
http://oeis.org/A108506 Numbers n such that 10^n - 59 is a prime. [>40000]
10^n - 61 is always divisible by 3
http://oeis.org/A178433 Numbers n such that 10^n - 63 is a prime. [>100000]
10^n - 67 is always divisible by 3
http://oeis.org/A177866 Numbers n such that 10^n - 69 is a prime. [>100000]
http://oeis.org/A178434 Numbers n such that 10^n - 71 is a prime. [>100000]
10^n - 73 is always divisible by 3
http://oeis.org/A178436 Numbers n such that 10^n - 77 is a prime. [>100000]
10^n - 79 is always divisible by 3
http://oeis.org/A178437 Numbers n such that 10^n - 81 is a prime. [>100000]
http://oeis.org/A178438 Numbers n such that 10^n - 83 is a prime. [>100000]
http://oeis.org/A108331 Numbers n such that 10^n - 87 is a prime. [>15328]
http://oeis.org/A108332 Numbers n such that 10^n - 89 is a prime. [>40000]
10^n - 91 is always divisible by 3
http://oeis.org/A178531 Numbers n such that 10^n - 93 is a prime. [>100000]
10^n - 97 is always divisible by 3
http://oeis.org/A178439 Numbers n such that 10^n - 99 is a prime. [>100000]



http://www.worldofnumbers.com/deplat.htm#pdp101 Note on numbers of the form 10^n + 1.
http://oeis.org/A049054 Numbers n such that 10^n + 3 is a prime. [>100000]
http://oeis.org/A088274 Numbers n such that 10^n + 7 is a prime. [>100000]
http://oeis.org/A088275 Numbers n such that 10^n + 9 is a prime. [>100000]
10^n + 11 is always divisible by 3.
http://oeis.org/A095688 Numbers n such that 10^n + 13 is a prime. [>100000]
10^n + 17 is always divisible by 3
http://oeis.org/A108052 Numbers n such that 10^n + 19 is a prime. [>100000]
http://oeis.org/A108050 Numbers n such that 10^n + 21 is a prime. [>100000]
10^n + 23 is always divisible by 3
http://oeis.org/A108312 Numbers n such that 10^n + 27 is a prime. [>100000]
10^n + 29 is always divisible by 3
http://oeis.org/A107083 Numbers n such that 10^n + 31 is a prime. [>100000]
http://oeis.org/A107084 Numbers n such that 10^n + 33 is a prime. [>100000]
http://oeis.org/A135109 Numbers n such that 10^n + 37 is a prime. [>72631]
http://oeis.org/A135108 Numbers n such that 10^n + 39 is a prime. [>81778]
10^n + 41 is always divisible by 3
http://oeis.org/A108049 Numbers n such that 10^n + 43 is a prime. [>35393]
10^n + 47 is always divisible by 3
http://oeis.org/A108054 Numbers n such that 10^n + 49 is a prime. [>80883]
http://oeis.org/A135118 Numbers n such that 10^n + 51 is a prime. [>47026]
10^n + 53 is always divisible by 3
http://oeis.org/A135119 Numbers n such that 10^n + 57 is a prime. [>100000]
10^n + 59 is always divisible by 3
http://oeis.org/A135116 Numbers n such that 10^n + 61 is a prime. [>100000]
http://oeis.org/A135115 Numbers n such that 10^n + 63 is a prime. [>100000]
http://oeis.org/A135113 Numbers n such that 10^n + 67 is a prime. [>100000]
http://oeis.org/A135114 Numbers n such that 10^n + 69 is a prime. [>100000]
10^n + 71 is always divisible by 3
http://oeis.org/A135132 Numbers n such that 10^n + 73 is a prime. [>100000]
10^n + 77 is always divisible by 3
http://oeis.org/A135131 Numbers n such that 10^n + 79 is a prime. [>63916]
http://oeis.org/A137848 Numbers n such that 10^n + 81 is a prime. [>100000]
10^n + 83 is always divisible by 3
http://oeis.org/A135117 Numbers n such that 10^n + 87 is a prime. [>100000]
10^n + 89 is always divisible by 3
http://oeis.org/A110918 Numbers n such that 10^n + 91 is a prime. [>100000]
http://oeis.org/A135112 Numbers n such that 10^n + 93 is a prime. [>100000]
http://oeis.org/A135107 Numbers n such that 10^n + 97 is a prime. [>100000]
http://oeis.org/A110980 Numbers n such that 10^n + 99 is a prime. [>100000]



Many solutions above 10000 were found and published in :
http://www.primenumbers.net/prptop/prptop.php
by Milton L. Brown and are recapitulated in the list...
... but need be doublechecked !!!
as some are not the smallest of length x+1 or the largest of length x.
[This has been done in the mean time]

See also WONPLATE 161 at http://www.worldofnumbers.com/em161.htm

Sample PFGW code used

ABC2 10^$b-$a // {number_primes,$b,1}
a:from 1 to 1000001 step 2
b:from 10290 to 10299

Save this text file e.g. as 'border.txt' and
then execute the following command line :

pfgw -f border.txt

look in pfgw.log for found prp's
look in pfgw_err.log for errors that occurred (very rare though)

Reserved ranges

Patrick De Geest

nihil

Happy prp hunting...

Largest gaps sorted by merit > 10

How to calculate the merit of a gap ?

merit = gapsize / ( x * ln(10) )

meritgapxby
12,20739837014173MLB
11,23841786416148AN
11,14928679611172PDG
11,13864403225112PDG
10,93954003821440PDG
10,13445062819311PDG
10,13372572431103PDG
10,03643050018630PDG

Record prime gaps

Between 10^91230 – 911141 and 10^91230 + 7921 is a prime gap of 919062
Between 10^46843 – 405809 and 10^46843 + 483673 is a prime gap of 889482
Between 10^77418 – 33 and 10^77418 + 841683 is a prime gap of 841716
Between 10^47006 – 388097 and 10^47006 + 389383 is a prime gap of 777480
Between 10^999999 – 172473 and 10^999999 + 593499 is a prime gap of 765972
Between 10^31103 – 86991 and 10^31103 + 638733 is a prime gap of 725724 Note : its merit value is 10,133; the digit anagrams between 31103 and 10,133; 724 & 725 are consecutieve !
Between 10^35212 – 142491 and 10^35212 + 546891 is a prime gap of 689382

Record displacement value

10^91230 – 911141
10^77418 + 841683
10^31103 + 638733
10^35301 – 614489
10^35257 – 601269
10^35212 + 546891
About an older record displacement value
10^30999 + 349947
ps. displacement +y equals this palindromic expression 3 * 38883 * 3
The palindromic midfactor is further divisible by 3 & 13!
Am I lucky or am I unlucky ...

Curios

Here is a nice list of what I called efficient prp's
where the displacement equals the exponent.
Note that the below lists might be incomplete.

10^3 – 3
10^23 – 23
10^171 – 171
10^903 – 903
10^48107 – 48107
10^48449 – 48449

10^1 + 1
10^451 + 451
10^13807 + 13807
10^23769 + 23769


                    10^(x-1)                                                10^x
      x-1             B                          x                            B             x+1
 ————————————————————>O<—————————————————————————————————————————————————————>O<————————————————————
                      R                                                       R
 ---------------o-----D-----o-------------------------------------------o-----D-----o---------------
                      E                                                       E     
               -y     R    +y                                          -y     R    +y

It sometimes happens that the smallest and largest borderprp
of a certain length x have identical absolute displacement values.
Here is my list so far :
     
10^47 + 33 and 10^48 – 33
10^362 + 471 and 10^363 – 471
10^482 + 177 and 10^483 – 177
10^9841 + 657 and 10^9842 – 657

It sometimes happens that the negative and positive displacements
from a common border 10^x are the same i.e., equidistant.
Here is my list so far :
     
10^17 – 3 and 10^17 + 3
10^45 – 9 and 10^45 + 9
10^3175 – 13431 and 10^3175 + 13431

Same 'outerbounds'.
     
10^17 – 3 and 10^18 + 3
10^19 – 39 and 10^20 + 39
10^46 – 33 and 10^47 + 33
10^153 – 453 and 10^154 + 453
10^195 – 39 and 10^196 + 39
10^277 – 273 and 10^278 + 273
10^315 – 627 and 10^316 + 627
10^354 – 129 and 10^355 + 129
10^582 – 2277 and 10^583 + 2277
10^2024 – 6741 and 10^2025 + 6741
10^5500 – 1767 and 10^5501 + 1767

Identical displacements for consecutive values x

Three consecutives

10^1 – 3
10^2 – 3
10^3 – 3

10^0 + 1
10^1 + 1
10^2 + 1

10^65 + 49
10^66 + 49
10^67 + 49
Two consecutives

10^49 – 57
10^50 – 57

10^303 – 59
10^304 – 59

10^3144 – 531
10^3145 – 531

10^9823 – 39261
10^9824 – 39261

10^5 + 3
10^6 + 3

10^8 + 7
10^9 + 7

10^17 + 3
10^18 + 3

10^12990 + 11163
10^12991 + 11163

10^14872 + 27321
10^14873 + 27321


Messages, annotations, links, etc.

10^581 ± y curious 10^581 - 851 and 10^581 + 1581 !

10^663 ± y curious as only digits 3 & 6 occur ! 10^663 - 33 and 10^663 + 6333 and also the gap shares this property *6366.

10^999 ± y 10^999+7 and Titanix by MLB

10^3669 ± y displacement -y = 9663 is 'reverse' of exponent 3669.

10^5020 ± y Prime Gap of 82794 by MLB

10^5028 ± y Prime Gap of 48116 by MLB

10^9999 ± y 10^9999 + 33603 is genuine prime http://www.worldofnumbers.com/em144.htm

10^10000 ± y 101 PRP Gaps by MLB

10^10087 ± y commented by MLB " This is an interesting prime, n'est ce pas 10^10087+10089,10088 "

10^10003 ± y (Erroneous) Prime Gap of 72580 by MLB

10^10294 ± y 10^10294 + 20491 note that exponent and displacement are anagrams of each other.

10^10702 ± y (Alleged) Prime gap of 364,188 by MLB

10^10781 ± y 10^10781 + 781 : note the doubling of numbers when split like 10^10|781+781 !

10^12000 ± y 10^12000+3213 and P=/= NP by MLB

10^12002 ± y Prime Number GAP of 123,380 by MLB

10^12012 ± y (Wrong) Prime GAP of 61946 by MLB

10^12990 ± y 10^12990 + 11163 http://groups.yahoo.com/group/primenumbers/message/18227
10^12991 ± y 10^12991 + 11163 http://groups.yahoo.com/group/primenumbers/message/18229

10^13000 ± y 10^13000+10029 by MLB

10^14173 ± y Prime Gap of 398,370 by MLB and verified by TA

10^16000 ± y 10^16000+23967 is PRP (but also +7749 !) by MLB

10^18000 ± y Prime Gap of 69438 by MLB

10^20000 ± y Prime Gap of 118576 and 10^20000 + 56149 by MLB

10^31001 ± y 10^31001+336963 is PRP by MLB

10^117059 + 28879 is 3,5,7-PRP! and is the only one up to + 300000.

10^178695 + 136719 is 3,5,7-PRP! and is the only one up to + 300000.

10^180979 + 79477 is 3,5,7-PRP! The next one is 10^180979 + 107739.

10^181182 + 383973 is 3,5,7-PRP! and is the only one up to + 600000.

10^249010 + 348861 is 3,5,7-PRP! and is the only one up to + 600000.


Palindromes with at least 5 digits

2924		-6209		+32323		*38532		+ palindromic : 32323 is a SUPP
3059		-2531		+10801		*13332		+ palindromic
3497		-19071		+24763		*43834		* palindromic gap
4217		-14741		+11103		*25844		- palindromic
4224		-22949		+1293		*24242		* palindromic gap
4244		-15351		+15873		*31224		- palindromic
4264		-4157		+10701		*14858		+ palindromic
4279		-24033		+3039		*27072		* palindromic gap
4320		-17589		+10993		*28582		* palindromic gap
4461		-3833		+12921		*16754		+ palindromic
4512		-22229		+6553		*28782		* palindromic gap
4646		-1389		+20223		*21612		* palindromic gap
4882		-7559		+13431		*20990		+ palindromic
5022		-14571		+9061		*23632		* palindromic gap
5118		-13731		+7543		*21274		- palindromic
5166		-5367		+42607		*47974		* palindromic gap
5203		-12027		+11311		*23338		+ palindromic 
5206		-2337		+12721		*15058		+ palindromic
5227		-10001		+18373		*28374		- palindromic
5230		-8459		+15373		*23832		* palindromic gap
5253		-4251		+18171		*22422		* palindromic gap
5289		-38283		+919		*39202		- & + palindromic
5339		-7913		+10101		*18014		+ palindromic
5362		-14141		+15349		*29490		- palindromic
5458		-10901		+6427		*17328		- palindromic
5487		-15251		+13549		*28800		- palindromic
5505		-34097		+12421		*46518		+ palindromic
5631		-33933		+11343		*45276		- palindromic
5674		-21107		+35853		*56960		+ palindromic
5775		-31013		+1621		*32634		- palindromic
5894		-18441		+10401		*28842		+ palindromic
5926		-15651		+17761		*33412		- palindromic
6008		-36537		+12247		*48784		* palindromic gap
6011		-11211		+7897		*19108		- palindromic
6022		-11493		+9309		*20802		* palindromic gap
6062		-539		+27943		*28482		* palindromic gap
6088		-21251		+19291		*40542		+ palindromic
6101		-22691		+2761		*25452		* palindromic gap
6117		-10589		+11511		*22100		+ palindromic
6128		-24741		+12721		*37462		+ palindromic
6137		-6693		+15019		*21712		* palindromic gap
6175		-5489		+32023		*37512		+ palindromic
6181		-12039		+14041		*26080		+ palindromic
6278		-6753		+12721		*19474		+ palindromic
6358		-18581		+11277		*29858		- palindromic
6380		-17471		+24339		*41810		- palindromic
6414		-13631		+11911		*25542		- & + palindromic
6418		-13431		+6171		*19602		- palindromic
6480		-38283		+2007		*40290		- palindromic
6558		-21089		+38283		*59372		+ palindromic  
6559		-12021		+13951		*25972		- palindromic
6577		-17871		+7713		*25584		- palindromic
6633		-11811		+15073		*26884		- palindromic
6674		-1931		+18081		*20012		+ palindromic
6689		-11411		+1617		*13028		- palindromic
6725		-18381		+6181		*24562		- palindromic
6762		-90197		+31713		*121910		+ palindromic
6801		-21441		+5731		*27172		* palindromic gap
6811		-15449		+47277		*62726		* palindromic gap
6848		-18581		+13813		*32394		- palindromic
6855		-9737		+19291		*29028		+ palindromic
6923		-35147		+11211		*46358		+ palindromic
6931		-9423		+12721		*22144		+ palindromic
6951		-1821		+19291		*21112		+ palindromic & palindromic gap
6958		-12083		+11211		*23294		+ palindromic
6968		-10001		+6139		*16140		- palindromic
6974		-16761		+20191		*36952		- palindromic
7006		-15051		+4291		*19342		- palindromic
7069		-36957		+13231		*50188		+ palindromic
7088		-17571		+25479		*43050		- palindromic
7095		-10301		+2007		*12308		- palindromic
7164		-6531		+18921		*25452		* palindromic gap
7182		-30947		+19891		*50838		+ palindromic
7205		-1071		+18381		*19452		+ palindromic
7309		-11883		+10339		*22222		* palindromic gap (repdigital)
7385		-30003		+12901		*42904		- palindromic
7465		-36063		+5613		*41676		- palindromic
7586		-9261		+15991		*25252		* palindromic gap
7606		-9323		+33601		*42924		* palindromic gap
7615		-1997		+15351		*17348		+ palindromic
7657		-5313		+10101		*15414		+ palindromic
7690		-16161		+68511		*84672		- palindromic
7698		-14811		+9831		*24642		* palindromic gap
7725		-11711		+42367		*54078		- palindromic
7753		-3239		+17071		*20310		+ palindromic
7802		-29951		+10753		*40704		* palindromic gap
7855		-33059		+30303		*63362		+ palindromic
7907		-2151		+79467		*81618		* palindromic gap
7922		-31113		+13933		*45046		- palindromic
7924		-30369		+16461		*46830		+ palindromic
8002		-19353		+4479		*23832		* palindromic gap
8122		-3743		+16759		*20502		* palindromic gap
8128		-9563		+34171		*43734		* palindromic gap
8164		-27317		+13731		*41048		+ palindromic
8208		-19091		+6367		*25458		- palindromic
8209		-17171		+52881		*70052		- palindromic
8265		-39693		+25687		*65380		- palindromic
8292		-97079		+1027		*98106		- palindromic
8380		-21941		+1591		*23532		* palindromic gap
8479		-30903		+7771		38674		- palindromic
8525		-22233		+15751		*37984		+ palindromic
8533		-16847		+19291		*36138		+ palindromic
8653		-81		+13431		*13512		+ palindromic
8710		-8741		+17011		*25752		* palindromic gap
8714		-16661		+31401		*48062		- palindromic
8722		-15153		+29491		*44644		* palindromic gap
8817		-14399		+8323		*22722		* palindromic gap
8950		-17853		+47203		*65056		* palindromic gap
8953		-52047		+11811		*63858		+ palindromic
8995		-3963		+30903		*34866		+ palindromic
9005		-14441		+3969		*18410		- palindromic
9023		-14597		+36463		*51060		+ palindromic
9031		-22667		+32923		*55590		+ palindromic
9055		-30231		+14313		*44544		* palindromic gap
9067		-12507		+13231		*25738		+ palindromic
9081		-11711		+24387		*36098		- palindromic
9238		-37857		+13531		*51388		+ palindromic
9251		-11211		+5967		*17178		- palindromic
9254		-33503		+36063		*69566		+ palindromic
9295		-78587		+35829		*114416		- palindromic
9365		-14841		+51873		*66714		- palindromic
9408		-35753		+12387		*48140		- palindromic
9487		-16427		+30037		*46464		* palindromic gap
9518		-40901		+2533		*43434		* palindromic gap
9533		-1917		+70407		*72324		+ palindromic
9574		-17171		+37659		*54830		- palindromic
9636		-17771		+16017		*33788		- palindromic
9643		-6927		+30003		*36930		+ palindromic
9720		-34023		+6781		*40804		* palindromic gap
9739		-20049		+16561		*36610		+ palindromic
9763		-24897		+17127		*42024		* palindromic gap
9797		-4859		+14641		*19500		+ palindromic
9849		-12933		+31111		*44044		* palindromic gap
9894		-53189		+13431		*66620		+ palindromic
9906		-24341		+57687		*82028		* palindromic gap
9923		-33831		+12121		*45952		+ palindromic
10010		-28787		+32929		*61716		* palindromic gap
10013		-4383		+20059		*24442		* palindromic gap
10028		-16761		+109081		*125842		- palindromic
10096		-8489		+81009		*89498		* palindromic gap
10155		-77877		+471		*78348		- palindromic
10238		-21831		+10401		*32232		+ palindromic
10333		-1103		+70807		*71910		+ palindromic
10364		-17771		+49129		*66900		- palindromic
10398		-7791		+17151		*24942		* palindromic gap
10416		-25379		+39193		*64572		+ palindromic
10481		-57647		+9829		*67476		* palindromic gap
10499		-36009		+31213		*67222		+ palindromic
10541		-20001		+33333		*53334		+ palindromic
10624		-11303		+39193		*50496		+ palindromic
10642		-12021		+13521		*25542		- palindromic
10674		-39713		+11011		*50724		+ palindromic
10726		-2073		+99799		*101872		+ palindromic
10728		-12371		+7731		*20102		* palindromic gap
10878		-32123		+14551		*46674		- palindromic
10918		-17019		+4593		*21612		* palindromic gap
10936		-17771		+98331		*116102		- palindromic
10965		-21347		+42489		*63836		* palindromic gap
10982		-7773		+60313		*68086		* palindromic gap
11057		-30863		+32673		*63536		* palindromic gap
11112		-37673		+1197		*38870		- palindromic
11142		-37373		++12493		*49866		- palindromic
11148		-23517		+17287		*40804		* palindromic gap
11198		-6399		+30703		*37102		+ palindromic
11205		-10301		+8943		*19244		- palindromic
11214		-18711		+17971		*36682		+ palindromic
11311		-20447		+25107		*45554		* palindromic gap (as well as the exponent !)
11313		-4631		+10801		*15432		+ palindromic
11340		-31749		+13431		*45180		+ palindromic
11343		-17631		+11611		*29242		+ palindromic
11544		-31341		+18781		*50122		+ palindromic
11624		-38183		+5281		*43464		- palindromic
11658		-10239		+32323		*42562		+ palindromic : 32323 is a SUPP
11811		-1071		+32923		*33994		+ palindromic (as well as the exponent !)
11834		-18281		+4323		*22604		- palindromic
11916		-1511		+43743		*45254		* palindromic gap
11968		-11711		+25341		*37052		- palindromic
11983		-29241		+10801		*40042		+ palindromic
12022		-38471		+10113		*48584		* palindromic gap
12121		-14441		+2947		*17388		- palindromic (as well as the exponent !)
12050		-2409		+70507		*72916		+ palindromic
12097		-7343		+17809		*25152		* palindromic gap
12116		-37457		+12337		*49794		* palindromic gap
12166		-23427		+63751		*87178		* palindromic gap
12188		-36963		+57843		*94806		- palindromic
12252		-13731		+15601		*29332		- palindromic
12262		-26309		+34197		*60506		* palindromic gap
12289		-29289		+17871		*47160		+ palindromic
12371		-53907		+15589		*69496		* palindromic gap
12379		-16661		+22563		*39224		- palindromic
12443		-9309		+36663		*45972		+ palindromic
12471		-2679		+75157		*77836		+ palindromic
12579		-38283		+18837		*57120		- palindromic
12599		-13931		+79089		*93020		- palindromic
12612		-4793		+22279		*27072		* palindromic gap
12621		-17559		+17871		*35430		+ palindromic (as well as the exponent !)
12677		-10401		+21909		*32310		- palindromic
12681		-7911		+32793		*40704		* palindromic gap
12725		-12933		+39393		*52326		+ palindromic
12739		-19191		+54361		*73552		- palindromic
12766		-7721		+14191		*21912		* palindromic gap
12786		-18581		+7663		*26244		- palindromic
12801		-16461		+38323		*54784		- palindromic
12815		-57233		+19191		*76424		+ palindromic
12854		-91203		+11311		*102514		+ palindromic
12881		-7889		+58777		*66666		* palindromic gap (repdigital)
12885		-87039		+10101		*97140		+ palindromic
12901		-10001		+33057		*43058		- palindromic
12919		-96069		+19857		*115926		- palindromic
12924		-39693		+30691		*70384		- palindromic
12982		-10701		+9603		*20304		- palindromic
13068		-15651		+198723		*214374		- palindromic
13120		-13173		+32223		*45396		+ palindromic
13136		-80721		+7057		*87778		* palindromic gap
13235		-25901		+18843		*44744		* palindromic gap
13264		-35153		+14287		*49440		- palindromic
13269		-2819		+22833		*25652		* palindromic gap
13275		-15251		+15507		*30758		- palindromic
13285		-30003		+7561		*37564		- palindromic
13296		-5663		+18579		*24242		* palindromic gap
13367		-48723		+33633		*82356		+ palindromic
13412		-8877		+11911		*20788		+ palindromic
13535		-31313		+2407		*33720		- palindromic
13598		-13307		+11511		*24818		+ palindromic
13601		-23279		+31413		*54692		+ palindromic
13606		-102821		+13731		*116552		+ palindromic
13660		-19673		+43153		*62826		* palindromic gap
13708		-77877		+10623		*88500		- palindromic
13717		-63809		+14341		*78150		+ palindromic
13860		-12779		+11611		*24390		+ palindromic
13899		-3201		+11011		*14212		+ palindromic
13916		-86891		+987	   	*87878		* palindromic gap
14000		-13587		+27927		*41514		* palindromic gap
14003		-16739		+8103		*24842		* palindromic gap
14019		-9339		+171	   	*9510		- & + palindromic
14040		-11937		+52309	   	*64246		* palindromic gap
14103		-12321		+4683		*17004		- palindromic
14156		-34643		+17443	   	*52086		- palindromic
14206		-11009		+30003		*41012		+ palindromic
14272		-2361		+74947		*77308		+ palindromic
14385		-14441		+1113		*15554		- palindromic
14403		-15251		+25021	   	*40272		- palindromic
14454		-13853		+34743	   	*48596		+ palindromic
14469		-13931		+7521	   	*21452		- palindromic
14490		-39741		+7833	   	*47574		* palindromic gap
14554		-95759		+55773	   	*151532		- palindromic
14596		-15581		+18181	   	*33762		+ palindromic
14689		-95297		+12421	   	*107718		+ palindromic
14784		-20919		+5743	   	*26662		* palindromic gap
14806		-23997		+23877		*47874		* palindromic gap
14828		-90809		+79669		*170478		- palindromic
14956		-19469		+5683		*25152		* palindromic gap
14994		-113379		+98689		*212068		+ palindromic
15013		-73757		+12001		*85758		* palindromic gap
15047		-190241		+91519		*281760		+ palindromic
15078		-15981		+33913		*49894		* palindromic gap
15116		-3419		+18393		*21812		* palindromic gap
15118		-7241		+10701		*17942		+ palindromic
15143		-13839		+34543		*48382		+ palindromic
15202		-37073		+363		*37436		- & + palindromic
15266		-3399		+99199		*102598		+ palindromic
15268		-10601		+64801		*75402		- palindromic
15402		-38283		+4827		*43110		- palindromic
15412		-14241		+123447		*137688		- palindromic
15431		-23619		+14841		*38460		+ palindromic
15496		-32279		+16461		*48740		+ palindromic
15519		-14697		+27927		*42624		* palindromic gap
15541		-75609		+98989		*174598		+ palindromic
15567		-18293		+45853		*64146		* palindromic gap
15590		-9443		+17971		*27414		+ palindromic
15602		-75999		+33433		*109432		+ palindromic
15623		-6869		+15043		*21912		* palindromic gap
15645		-18789		+32623		*51412		+ palindromic
15723		-14841		+54169		*69010		- palindromic
15733		-25653		+31213		*56866		+ palindromic
15786		-23723		+36283		*60006		* palindromic gap
15816		-1713		+61323		*63036		* palindromic gap
15845		-51		+76167		*76218		+ palindromic
15882		-14441		+33337		*47778		- palindromic
15893		-23249		+3823		*27072		* palindromic gap
15953		-1661		+73137		*74798		+ palindromic
15993		-11211		+9669		*20880		- & + palindromic
15999		-35753		+9139		*44892		- palindromic
16068		-36701		+48147		*84848		* palindromic gap
16096		-33393		+11611		*45004		+ palindromic
16223		-31571		+10801		*42372		+ palindromic
16235		-24131		+92629		*116760		+ palindromic
16273		-124019		+12421		*136440		+ palindromic
16277		-26153		+77577		*103730		+ palindromic
16294		-48917		+16861		*65778		+ palindromic
16318		-9147		+13831		*22978		+ palindromic
16323		-19791		+16209		*36000		- palindromic
16337		-58599		+70407		*129006		+ palindromic
16420		-15351		+16779		*32130		- palindromic
16483		-12459		+74719		*87178		* palindromic gap
16492		-23891		+15451		*39342		+ palindromic
16624		-55307		+79197		*134504		+ palindromic
16676		-34779		+33433		*68212		+ palindromic
16686		-74693		+38083		*112776		+ palindromic
16740		-80597		+14641		*95238		+ palindromic
16796		-27509		+35527		*63036		* palindromic gap
16817		-38699		+71317		*110016		+ palindromic
16876		-8381		+11211		*19592		+ palindromic
16913		-13131		+15399		*28530		- palindromic
16966		-3269		+24603		*27872		* palindromic gap
16973		-12821		+351		*13172		- palindromic
17039		-43943		+70207		*114150		+ palindromic
17041		-27717		+58551		*86268		* palindromic gap
17139		-20241		+6831		*27072		* palindromic gap
17145		-17043		+24981		*42024		* palindromic gap
17151		-55017		+39793		*94810		+ palindromic
17169		-2037		+81501		*83538		* palindromic gap
17220		-39693		+64867		*104560		- palindromic
17254		-3041		+39993		*43034		+ palindromic & palindromic gap
17267		-12921		+13959		*26880		- palindromic
17278		-31613		+4383		*35996		- palindromic
17293		-36087		+161161		*197248		+ palindromic
17339		-21371		+3171		*24542		* palindromic gap
17361		-12321		+9237		*21558		- palindromic
17390		-42381		+92029		*134410		+ palindromic
17391		-17427		+23077		*40504		* palindromic gap
17424		-42993		+19891		*62884		+ palindromic
17461		-33233		+109227		*142460		- palindromic
17496		-753		+17671		*18424		+ palindromic
17532		-132821		+72727		*205548		+ palindromic : 72727 is a SUPP
17538		-6767		+19791		*26558		+ palindromic
17541		-11511		+102333		*113844		- palindromic
17574		-30903		+7987		*38890		- palindromic
17609		-8693		+11311		*20004		+ palindromic
17695		-55613		+7113		*62726		* palindromic gap
17714		-23477		+26217		*49694		* palindromic gap
17906		-10601		+35031		*45632		- palindromic
17984		-33633		+33823		*67456		- palindromic
18019		-76767		+17487		*94254		- palindromic
18027		-15449		+65769		*81218		* palindromic gap
18053		-10401		+67567		*77968		- palindromic
18086		-31113		+40191		*71304		- palindromic
18126		-11237		+14941		*26178		+ palindromic
18210		-80399		+32323		*112722		+ palindromic : 32323 is a SUPP
18239		-68207		+12001		*80208		* palindromic gap
18303		-36353		+9501		*45854		* palindromic gap
18312		-13031		+144567		*157598		- palindromic
18399		-11211		+78493		*89704		- palindromic
18406		-34341		+33661		*68002		- palindromic
18415		-28413		+39693		*68106		+ palindromic
18417		-24563		+21501		*46064		* palindromic gap
18436		-31313		+75961		*107274		- palindromic
18489		-11111		+96369		*107480		- & + palindromic
18497		-211961		+36363		*248324		+ palindromic
18506		-1883		+46801		*48684		* palindromic gap
18535		-41901		+42847		*84748		* palindromic gap
18537		-18881		+88959		*107840		- palindromic
18550		-6137		+75957		*82094		+ palindromic
18557		-34743		+108759		*143502		- palindromic
18601		-50759		+36363		*87122		+ palindomic
18636		-80283		+74847		*155130		+ palindomic
18657		-72627		+133417		*206044		- palindromic
18674		-18981		+73087		*92068		- palindromic
18695		-12821		+12429		*25250		- palindromic
18701		-48983		+15763		*64746		* palindromic gap
18722		-93839		+32469		*126308		- palindromic
18757		-31313		+21001		*52314		- palindromic
18758		-34341		+33469		*67810		- palindromic
18868		-15851		+3841		*19692		- palindromic
18985		-10101		+8149		*18250		- palindromic
19140		-8883		+79797		*88680		+ palindromic
19169		-91119		+69003		*160122		- palindromic
19200		-38883		+16161		*55044		- & + palindromic
19324		-30203		+8041		*38244		- palindromic
19340		-31913		+24481		*56394		- palindromic
19350		-10101		+26281		*36382		- palindromic
19357		-18081		+72141		*90222		- palindromic
19524		-25071		+37573		*62644		+ palindromic
19537		-17271		+51673		*68944		- palindromic
19591		-3483		+59953		*63436		* palindromic gap
19606		-75957		+32689		*108646		- palindromic
19629		-13521		+35353		*48874		+ palindromic prime
19864		-28521		+11611		*40132		+ palindromic
19868		-70007		+110071		*180078		- palindromic
19873		-36311		+17871		*54182		+ palindromic
19914		-10701		+6477		*17178		- palindromic
19915		-26877		+17071		*43948		+ palindromic
20001		-30863		+95359		*126222		+ palindromic
20007		-16929		+3273		*20202		* palindromic gap
20018		-72827		+46239		*119066		- palindromic
20045		-12321		+31213		*43534		-, + & * are all palindromic !
20080		-47021		+74347		*121368		+ palindromic
20111		-75557		+20691		*96248		- palindromic
20179		-33933		+149703		*183636		- palindromic
20183		-113957		+30103		*144060		+ palindromic
20333		-175571		+46843		*222414		- palindromic
20356		-93233		+74547		*167780		+ palindromic
20416		-31613		+81109		*112722		- palindromic
20437		-16661		+15483		*32144		- palindromic
20485		-94959		+18181		*113140		+ palindromic
20529		-11333		+28671		*40004		* palindromic gap
20606		-14141		+14461		*28602		- palindromic
20629		-13169		+11473		*24642		* palindromic gap
20641		-9977		+18681		*28658		+ palindromic
20705		-214143		+14241		*228384		+ palindromic
20641		-9977		+18681		*28658		+ palindromic
20734		-10731		+10381		*21112		* palindromic gap
20910		-34617		+17271		*51888		+ palindromic
20959		-13103		+10629		*23732		* palindromic gap
20976		-72327		+22251		*94578		- palindromic
20980		-37707		+32923		*70630		+ palindromic
21053		-55587		+12121		*67708		+ palindromic
21059		-31013		+68347		*99360		- palindromic
21073		-17549		+34243		*51792		+ palindromic
21076		-187887		+15451		*203338		+ palindromic
21087		-15551		+12621		*28172		- & + palindromic
21094		-18581		+1083		*19664		- palindromic
21117		-28331		+18633		*46964		* palindromic gap
21158		-7469		+12733		*20202		* palindromic gap
21219		-15737		+19591		*35328		+ palindromic
21340		-8111		+32923		*41034		+ palindromic
21359		-52551		+12421		*64972		+ palindromic
21366		-16581		+15451		*32032		+ palindromic
21401		-31913		+49647		*81560		- palindromic
21425		-14153		+26961		*41114		* palindromic gap
21431		-67809		+19969		*87778		* palindromic gap
21494		-4493		+20559		*25052		* palindromic gap
21581		-72327		+223681		*296008		- palindromic
21585		-10301		+16777		*27078		- palindromic
21651		-18987		+32323		*51310		+ palindromic : 32323 is a SUPP
21660		-3489		+37573		*41062		+ palindromic
21709		-34743		+101437		*136180		- palindromic
21716		-25821		+71017		*96838		+ palindromic
21752		-16073		+51703		*67776		* palindromic gap
21803		-8891		+16761		*25652		+ palindromic & palindromic gap
21848		-32823		+32839		*65662		- palindromic
21869		-97779		+54747		*152526		- palindromic
21874		-9803		+176671		*186474		+ palindromic
21896		-59517		+78787		*138304		+ palindromic : 78787 is a SUPP
21945		-130743		+12721		*143464		+ palindromic
21962		-207123		+32809		*239932		* palindromic gap
21969		-25803		+37123		*62926		* palindromic gap
22099		-95121		+11611		*106732		+ palindromic
22163		-12093		+110011		*122104		+ record palindromic
22267		-19391		+36531		*55922		- palindromic
22304		-90209		+78001		*168210		- palindromic
22433		-134289		+16761		*151050		+ palindromic
22435		-14183		+13389		*27572		* palindromic gap
22440		-72501		+10837		*83338		* palindromic gap
22494		-33533		+1899		*35432		- palindromic
22524		-44921		+11611		*56532		+ palindromic
22673		-22473		+1159		*23632		* palindromic gap
22785		-39893		+4653		*44546		- palindromic
22884		-36641		+3963		*40604		* palindromic gap
22905		-97079		+29467		*126546		- palindromic
22955		-38583		+4999		*43582		- palindromic
23006		-23709		+12621		*36330		+ palindromic
23026		-31613		+16329		*47942		- palindromic
23100		-34443		+43863		*78306		- palindromic
23126		-11589		+74547		*86136		+ palindromic
23181		-4539		+15351		*19890		+ palindromic
23217		-77577		+76479		*154056		- palindromic
23223		-79467		+71217		*150684		+ palindromic
23315		-50469		+12421		*62890		+ palindromic
23317		-12431		+33123		*45554		* palindromic gap
23343		-19791		+46441		*66232		- palindromic
23344		-52557		+98289		*150846		+ palindromic	(consecutive to 23343)
23351		-63077		+32623		*95700		+ palindromic
23365		-13631		+30021		*43652		- palindromic
23380		-23961		+39493		*63454		+ palindromic
23416		-52739		+17871		*70610		+ palindromic
23457		-77633		+97779		*175412		+ palindromic
23480		-112427		+76567		*188994		+ palindromic
23487		-28659		+18181		*46840		+ palindromic
23538		-17171		+86563		*103734		- palindromic
23545		-22143		+6439		*28582		* palindromic gap
23562		-15417		+49429		*64846		* palindromic gap
23648		-94449		+62349		*156798		- palindromic
23654		-23769		+94849		*118618		+ palindromic
23678		-73337		+112323		*185660		- palindromic
23697		-26469		+92629		*119098		+ palindromic
23776		-4703		+22369		*27072		* palindromic gap
23793		-34043		+28509		*62552		- palindromic
23806		-149171		+78687		*227858		+ palindromic
23849		-124421		+55333		*179754		- palindromic
23904		-40583		+71617		*112200		+ palindromic
23916		-5891		+18781		*24672		+ palindromic
23993		-37473		+24979		*62452		+ palindromic
24049		-131151		+75057		*206208		+ palindromic
24105		-34043		+70453		*104496		- palindromic
24138		-4491		+43993		*48484		* palindromic gap
24210		-177329		+31113		*208442		+ palindromic
24216		-30903		+81891		*112794		- palindromic
24237		-72627		+13737		*86364		- palindromic
24263		-38583		+13167		*51750		- palindromic
24269		-107577		+77377		*184954		+ palindromic
24335		-597		+42837		*43434		* palindromic gap
24346		-43983		+18381		*62364		+ palindromic
24351		-15951		+921		*16872		- palindromic
24381		-19791		+48789		*68580		- palindromic
24386		-26309		+40357		*66666		* palindromic gap
24405		-39039		+78787		*117826		+ palindromic
24431		-15827		+33333		*49160		+ palindromic
24451		-34071		+94449		*128520		+ palindromic
24453		-31013		+20143		*51156		- palindromic
24499		-10001		+87379		*97380		- palindromic
24515		-103301		+100017		*203318		- palindromic
24599		-93731		+70707		*164438		+ palindromic
24712		-26567		+17067		*43634		* palindromic gap
24747		-70007		+51399		*121406		- palindromic
24801		-20549		+63189		*83738		* palindromic gap
24812		-26783		+34533		*61316		* palindromic gap
24879		-94047		+10801		*104848		+ palindromic
24885		-33413		+13431		*46844		+ palindromic
24994		-117711		+84007		*201718		- palindromic
25004		-12267		+76021		*88288		* palindromic gap
25022		-96569		+62601		*159170		- palindromic
25040		-2541		+16561		*19102		+ palindromic
25154		-15851		+86929		*102780		- palindromic
25178		-71427		+9381		*80808		* palindromic gap
25185		-15219		+38583		*53802		+ palindromic
25187		-13163		+13299		*26462		* palindromic gap
25194		-4059		+60487		*64546		* palindromic gap
25271		-189951		+18981		*208932		+ palindromic
25370		-64979		+12321		*77300		+ palindromic
25375		-16979		+3723		*20702		* palindromic gap
25434		-39993		+2581		*42574		- palindromic
25477		-38283		+82887		*121170		- palindromic
25588		-73523		+15651		*89174		+ palindromic
25608		-27341		+19323		*46664		* palindromic gap
25656		-5249		+78289		*83538		* palindromic gap
25658		-121121		+43663		*164784		- palindromic
25750		-40493		+5671		*46164		* palindromic gap
25767		-97379		+439		*97818		- palindromic
25781		-10331		+19591		*29922		+ palindromic
25898		-12213		+11511		*23724		+ palindromic
25906		-14801		+12621		*27422		+ palindromic
25936		-10401		+129351		*139752		- palindromic
26030		-18281		+88369		*106650		- palindromic
26125		-17871		+32611		*50482		- palindromic
26166		-14861		+76267		*91128		+ palindromic
26239		-79797		+50857		*130654		- palindromic
26259		-13109		+15673		*28782		* palindromic gap
26297		-20631		+12621		*33252		+ palindromic
26300		-25809		+38737		*64546		* palindromic gap
26377		-26973		+12921		*39894		+ palindromic
26396		-54849		+122221		*177070		+ palindromic
26412		-34943		+79321		*114264		- palindromic
26484		-61949		+17571		*79520		+ palindromic
26538		-11811		+17997		*29808		- palindromic
26561		-17199		+37873		*55072		+ palindromic
26573		-97779		+14967		*112746		- palindromic
26594		-38583		+2029		*40612		- palindromic
26639		-70781		+36163		*106944		+ palindromic
26653		-17171		+86851		*104022		- palindromic
26657		-99399		+37017		*136416		- palindromic
26771		-35339		+96069		*131408		+ palindromic
26836		-124421		+5583		*130004		- palindromic
26849		-57131		+25297		*82428		* palindromic gap
27033		-16061		+1507		*17568		- palindromic


27200		-17591		+70207		*87798		+ palindromic
27400		-80757		+91119		*171876		+ palindromic
27729		-38883		+49467		*88350		- palindromic
27777		-316613		+64111		*380724		- palindromic
28814		-15651		+18873		*34524		- palindromic
29467		-4841		+77787		*82628		* palindromic gap
29599		-15251		+106231		*121482		- palindromic
30001		-19691		+93123		*112814		- palindromic
30008		-97979		+24019		*121998		- palindromic
30013		-36063		+64447		*100510		- palindromic
30200		-62529		+19089		*81618		* palindromic gap
31561		-52169		+15651		*67820		+ palindromic
33700		-224091		+35553		*259644		+ palindromic
35045		-201141		+93339		*294480		+ palindromic
35050		-379973		+66481		*446454		- palindromic
35112		-41127		+91819		*132946		+ palindromic
35299		-39957		+72927		*112884		+ palindromic
36569		-3741		+79087		*82828		* palindromic gap
36700		-37281		+19891		*57172		+ palindromic
37077		-23573		+75757		*99330		+ palindromic
42674		-40677		+10501		*51178		+ palindromic
48442		-33533		+57		*33590		- palindromic
96377		-20001		+199911		*219912		* palindromic gap


Repunits and repdigits with at least 3 digits

111		55-, 67-, 460+, 784+, 3248+, 5194+, 12119-,
222		88*,
333		222-, 231+, 1130-, 2111+, 7046+, 7110+, 13391+,
444		51*, 703*,
666		2300*,			* Number of the Beast. Beastly gap !
777		461+, 695-, 1013+, 1018-, 1289-, 1755-, 2961+, 5993-, 11801+, 17343+, 20766+, 23465+,
888		384*, 963*, 1685*,
999		183+, 1077-, 2104+, 2946-, 4046-, 4430-, 10803+,
1111		731+, 1512+, 1787+, 2146+, 3987+, 4056+, 5296+,
2222		1000*, 2879*,
3333		3197+, 4213+, 7042+, 8914-, 15485+,
7777		12459+, 17900+,
9999		5593+, 10525+, 17106-,
11111		18489-,
22222		7309*,
33333		9992-, 10541+, 24431+
66666		12881*, 24386*,

Tautonyms

1010		998*,
1212		665*, 2008*, 4732*,
1313		1622-, 3848-, 3933-, 5528-, 17552-, 19298-, 28399-, 
1414		4752*,
1616		16195*,
1717		2204+, 3922+, 8150+,
1818		643*, 1069*, 1155*, 1688*,
1919		351-, 1796-, 2977-, 4887-, 8239-, 11254-, 11663-, 13779-, 20411-,
2121		1832+, 2282+, 3058-, 4597-, 4658+, 4841-, 5623+, 8784+, 16656-, 24402-,
2323		518+, 638+, 1286+, 2360+, 3574+, 4970+, 9280+, 10766+, 11763+, 21793+, 25732+,
2424		601*,
2727		3359+, 3501+, 4773+, 7259+,
2828		379*,
2929		1563+, 2052+, 4497+, 5568+,
3131		1075-, 2158-, 4697-, 5545-, 5622-, 5765-, 15736-, 16440-, 20353-,
3232		625*,
3434		690*, 1939*,
3636		1246*,
3737		7420-, 27032-,
3838		2289*,
3939		1498+, 12918+, 21435-, 24525-, 26604+,
4040		721*, 6231*,
4141		1445+, 1533+, 10492+, 13774+,
4242		2476*, 4492*,
4343		2746-, 4664-, 9369-,
4747		11137+, 15427+,
4848		574*, 1080*, 6218*,
5151		1925+, 5087+, 5360-, 18037-, 18650-, 25514-,
5252		4567*,
5353		3131+,
5656		7072*,
5757		2062-, 3162-, 4478-, 4587+, 8130-, 15560-, 18650+, 20025-,
5959		1472+, 1903+, 6934+,
6060		18213*,
6363		793-, 1756+, 2599+, 2966-, 4439+, 6522-, 15045-, 18355-,
6767		4298-, 9159-, 17538-,
6969		1623-, 3326-, 4052+, 7553+, 12953+, 22166-, 26229-, 31285-,
7171		3153+,
7373		15234-,
7878		19414*,
7979		5196-,
8080		17594*,
8181		6586-, 10554+, 14263-, 17790+, 18367+, 19822+, 22350+,
8282		2652*, 4548*,
8383		10896+, 12172+, 20310+,
8484		2393*, 9439*, 23721*,
8686		3364*, 3765*,
8787		5668+, 6807+,
8989		5792+, 11447+, 14938+, 17531+,
9090		2724*,
9191		14568-, 15674-,
9292		2097*,
9393		9395+, 14397+, 17502-, 20515-, 24650-,
9898		3992*,
103103		42503-,
104104		12207*,
121121		25658-,
122122		25800*,
123123		23511-,
124124		21771*,
161161		17293+,
183183		23989-,
186186		20562*,
204204		36999*,
239239		35630+,
264264		15525*,

Varia & Curios

51		-323		+121		*444		- + * all palindromes
87		-273		+373		*646		+ * palindromic and 10^87+373 of palindromic length 88
135		-1463		+1143		*2606		first time both > 1000
180		-2183		+313		*2496		+ palindromic and 10^180+313 of palindromic length 181
384		-717		+171		*888		- + * all palindromes
387		-2909		+4449		*7358		* record value
568		-801		+501		*1302		* Slag van Vlaanderen (Battle of Flanders)
594		-11		+789		*800		* identical consecutive gaps see next
595		-503		+297		*800		* identical consecutive gaps see previous
756		-1713		+1711		*3424		near_equidistant displacements
787		-14397		+3171		*17568		- first time > 10000
844		-1241		+177		*1418		* first world war
967		-1269		+687		*1956		* PDG's birthyear 1956 is not that dull :-)
999		-6101		+7		*6108		+7 is proven prime by Titanix
1001		-3069		+9337		*12406		- came across repeat pattern '3069' in square !
1247		-10283		+10111		*20394		first time both > 10000
1956		-9459		+541		*10000		* PDG's birthyear 1956 gives a round gap of 10000 !
2742		-363		+6637		*7000		* rounded gap
3531		-7029		+21549		*28578		10^x ± y with range x from 3531 to 4201 by Dirk Augustin 5 okt 2006
3669		-9663		+4059		*13722		displacement is reverse of exponent !
4202		-11361		+14913		*26274		10^x ± y with range x from 4202 to 4999 by Dirk Augustin 27 dec 2006
5500		-1767		+18373		*20140		10^x ± y with range x from 5500 to 6300 by Dirk Augustin 13 maa 2007
5775		-31013		+1621		*32634		exponent and - are palindromic
6043		-15119		+4881		*20000		* rounded gap
6464		-20691		+2691		*23382		- is expansion of + : 2691 and 2{0}691
6678		-9081		+919		*10000		* a second round gap of 10000 like with exponent 1956
7000		-20181		+4981		*25162		10^x ± y with range x from 7000 to 7200 by Dirk Augustin 13 maa 2007
7095		-10301		+2007		*12308		palindromic - found by DA in the year +
7265		-6789		+18091		*24880		- ascending digits
7327		-13041		+10431		*23472		displacements are anagrams of each other : -13041 vs. +10431
7386		-1463		+14863		*16326		+ is expansion of - : 1463 and 14{8}63
9999		-11333		+33603		*44936		+33603 is genuine prime (next prp = +55377)
10002		-62057		+7341		*69398		DA (- 11 dec 2006 ) > -96191 by mlb
10003		-9489		+48967		*58456		DA (+ 03 aug 2006) < +63091 by mlb
10006		-7301		+4659		*11960		DA (- 12 dec 2006) > -7329 by mlb
10007		-19833		+9001		*28834		DA (- 12 dec 2006) > -22743 by mlb
10009		-3789		+45159		*48948		DA (+ 03 aug 2006) < +49521 by mlb
10010		-28787		+32929		*61716		DA (+ 03 aug 2006) < +59689 by mlb
10012		-1977		+44679		*46656		DA (+ 8 aug 2006) < +170781 by mlb
10013		-4383		+20059		*24442		DA (+ 5 aug 2006) < +29247 by mlb
10015		-17949		+36783		*54732		DA (+ 8 aug 2006) < +116961 by mlb
10020		-237		+16657		*16894		DA (+ 15 sep 2006) < +46291 by mlb
10022		-6569		+18843		*25412		DA (+ 15 sep 2006) < +52783 by mlb
10023		-31781		+6537		*38318		DA (+ 4 aug 2006) < +58333 by mlb
10025		-7067		+26323		*33390		DA (+ 8 aug 2006) < +105417 by mlb
10027		-29037		+49707		*78744		DA (+ 8 aug 2006) < +104749 by mlb
10028		-16761		+41911		*58672		DA (+ 8 aug 2006) < +109081 by mlb
10029		-3693		+62047		*65740		DA (+ 9 aug 2006) < +106123 by mlb
10030		-8951		+19713		*28664		DA (+ 7 aug 2006) < +101157 by mlb
10033		-2211		+30459		*32670		DA (+ 8 aug 2006) < +115821 by mlb
10096		-8489		+22533		*31022		DA (+ 8 aug 2006) < +81009 by mlb
10165		-6543		+2823		*9366		- descending digits
10172		-34697		+7711		*42408		PDG (- 8 jun 2007) > -43733 by mlb
10174		-33003		+7861		*40864		PDG (- 8 jun 2007) > -139377 by mlb
10181		-33797		+23329		*57126		PDG (- 10 jun 2007) > -99057 by mlb
10211		-18903		+65029		*83932		PDG (- 19 jun 2007) > -25521 by mlb
10213		-33917		+61327		*95244		PDG (- 19 jun 2007) > -66773 by mlb
10218		-48807		+7263		*56070		PDG (- 20 jun 2007) > -50729 by mlb
10222		-82437		+5781		*88218		PDG (- 21 jun 2007) > -103017 by mlb
10224		-9207		+7957		*17164		PDG (- 21 jun 2007) > -52301 by mlb
10225		-18317		+10009		*28326		PDG (- 21 jun 2007) > -63893 by mlb
10259		-18689		+18691		*37380		near equal equidistant displacements
10378		-5883		+8853		*14736		displacements are anagrams of each other : -5883 vs. +8853
10433		-21219		+957		*22176		PDG (- 17 sep 2007) > -21629 by mlb
10501		-21701		+30213		*56610		PDG (- 30 dec 2007) > -26397 by mlb
10508		-21491		+8253		*29744		PDG (- 5 jul 2008) > -86063 by mlb
10511		-7379		+23527		*30906		PDG (+ 3 jul 2008) < +24171 by mlb
10518		-44453		+19741		*64194		PDG (- 13 jul 2008) > -135993 by mlb
10538		-16707		+5193		*21900		PDG (- 17 jul 2008) > -27143 by mlb
10543		-73451		+77943		*151394		PDG (+ 13 dec 2006) < +233677 by mlb
10548		-44277		+5041		*49318		PDG (+ 29 jul 2008) < +14829 by mlb
10556		-25329		+21079		*46408		PDG (+ 29 jul 2008) < +62913 by mlb
10566		-76173		+24787		*100960		PDG (+ 30 jul 2008) < +26889 by mlb
10584		-39617		+26191		*65808		PDG (- 5 aug 2008) > -81617 by mlb
10615		-47787		+16543		*64330		PDG (- 6 aug 2008) > -89871 by mlb
10702		-108911		+19483		*128394		JKA (+ 12/2005 tbv) < +255277 by mlb
10704		-31563		+87601		*119164		PDG (+ 9 aug 2008) < +116707 by mlb
10847		-38597		+41403		*80000		* rounded gap
11114		-6957		+66667		*73624		PDG (+ 23 mei 2007) visually pleasing digits 10^11114+66667
11180		-8369		+21631		*30000		* rounded gap
11212		-13181		+44227		*57408		PDG (- 12 mei 2009) > -45117 by mlb
11909		-23841		+40011		*63852		PDG (+ 14 aug 2008) < +100537 by mlb
11921		-45681		+36187		*81868		PDG (+ 16 aug 2008) < +160917 by mlb
11956		-5817		+90883		*96700		PDG (+ 19 aug 2008) < +114139 by mlb
11959		-49457		+9133		*58590		PDG (+ 19 aug 2008) < +20841 by mlb
11966		-46437		+17781		*64218		PDG (+ 19 aug 2008) < +24871 by mlb
12002		-17189		+36823		*54012		PDG (+ 14 aug 2008) < +106191 by mlb
12007		-12707		+8239		*20946		PDG (- 26 apr 2008) > -20627 by mlb
12010		-14423		+14419		*28842		near equal displacements 14423 and 14419
12011		-26159		+10839		*36998		PDG (- 26 apr 2008) > -116069 by mlb
12012		-1559		+30879		*32438		DA (+ 2 aug 2006) < +60387 by mlb
12303		-32301		+54553		*86854		12303 is anagram of 32301
12332		-599		+13299		*13898		PDG (- 1 dec 2008) > -8781 by mlb
12346		-35183		+14253		*49436		PDG (- 30 aug 2009) > -86631 by mlb
12881		-7889		+58777		*66666		* palindromic gap (repdigital) a beastly gap !
12913		-120993		+5887		*126880		PDG (+ 22 mei 2007) < +178449 by mlb
12957		-21101		+15979		*37080		PDG (+ 12 dec 2008) < +97401 by mlb
12964		-7233		+36727		*43960		PDG (+ 13 dec 2008) < +95433 by mlb
12994		-293		+42259		*42552		PDG (+ 14 dec 2008) < +89241 by mlb
13310		-5691		+28081		*33772		PDG (+ 13 mei 2009) < +133963 by mlb
13704		-9279		+991		*10270		PDG (+ 13 nov 2009) < +101959 by mlb
13705		-28239		+1963 		*30202		PDG (+ 13 nov 2009) < +55887 by mlb
13714		-14547		+41371		*55918		41371 is anagram of 13714
14164		-7317		+30519		*37836		PDG (+ 7 dec 2009) < +59341 by mlb
14165		-31691		+21463		*53154		PDG (+ 7 dec 2009) < +30469 by mlb
14437		-1953		+1539	   	*3492		displacements are anagrams of each other : -1953 vs. +1539
15000		-8621		+76291		*84912		PDG (+ 17 dec 2006) < +114229 by mlb
15001		-7223		+62971		*70194		PDG (+ 14 dec 2009) < +212437 by mlb
15009		-65549		+10953		*76502		PDG (- 13 dec 2009) > -77151 by mlb
16000		-32961		+7749		*40710		DA (+ 2 aug 2006) < +23967 by mlb
16001		-4197		+3921		*8118		DA (+ 2 aug 2006) < +15757 by mlb
18464		-24207		+5793		*30000		* rounded gap
18650		-5151		+5757		*10908		-5151 and +5757 are two tautonyms on the same line
19664		-29433		+10567		*40000		* rounded gap
19776		-77699		+66799		*144498		surprising popup of three digits 6, 7 & 9
20001		-30863		+95359		*126222		PDG (+ 20 dec 2009) < +203889 by mlb
21224		-11193		+20539		*31732		PDG (- 24 feb 2010) > -76697 by mlb
22935		-29901		+32167		*62068		Gap *62068 is identical with next one
22936		-36867		+25201		*62068		Gap *62068 is identical with previous one
26003		-18911		+391		*19302		PDG (+ 15 feb 2010) < +26527 by mlb
27003		-18123		+116961		*135084		PDG (+ 16 apr 2008) < +126331 by mlb
27200		-17591		+70207		*87798		Use of the same three digits in 27200 and +70207
29020		-82719		+867		*83586		PDG (+ 8 maa 2008) < +29833 by mlb
35308		-131183		+27903		*159086		PDG (- 2 maa 2011) > -155591 by mlb
49999		-4149		+91701		*95850		PDG (+ started 11 feb 2005, ended 1 maa 2005)
59999		-516267		+65197		*581464		PDG (+ started 14 maa 2005, ended 30 maa 2005)
60959		-29177		+57		*29234		PDG (- 25 aug 2008) > -60959 by Henri Lifchitz
96377		-20001		+199911		*219912		First found palindromic gap with six digits 12 dec 2016

Largest palindromic displacements (update 10 nov 2017)
	10^35050 - 379973	
	10^27777 - 316613	
	10^21874 + 176671
	10^20333 - 175571
	10^17293 + 161161
	10^26836 - 124421
	10^26396 + 122221
	10^25658 - 121121
	10^24994 - 117711
	10^24515 - 103301



Noticing many numbers with displacement 10203 !
	10^2237 - 10203
	10^6100 + 10203
	10^7554 + 10203
	10^7805 + 10203
	10^11672 - 10203
	10^12037 - 10203
	10^21236 - 10203
	10^26645 + 10203

and their rearrangements like
	10^21095 + 20301
	10^7881 + 30201
	10^7687 with gap *30102

To do list

+/- 313		add to palprim4.htm
+/- 353		add to palprim4.htm
+/- 373		add to palprim4.htm
+/- 1001	add to wonplate 31
+/- 1111	add to wonplate 83
+/- 1617	add to consec.htm
+/- 1881	add to wonplate 22
+/- 3069	add to square.htm
+/- 6643	add to wonplate 98
+/- 10203	add to end of this page
+/- 1777771	add to wonplate 161











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E-mail address : pdg@worldofnumbers.com