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.
displacement +y < -y versus +y > -y (statistics)
Search for equal + and - values (equidistant).
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 could be obtained from [after an initial subscription]:
for Win32 could be obtained from:
and
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 http://groups.yahoo.com/group/primenumbers/,
the PrimeNumbers e-mail discussion list with message
http://groups.yahoo.com/group/primenumbers/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(25000-29999).txt borderprp(30000-34999).txt borderprp(35000-99999).txt |
| Two packed lists (in the form of PDF-files) of all borderprp's up to exponent 30000 were provided by Norman Luhn - [September 2018]. Exponent n and offset a (n_a) , where 10n – a is (probable) prime. Largest-n-digit-prp.pdfExponent n and offset a (n_a) , where 10n + a is (probable) prime. Smallest-n-digit-prp.pdf |
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>pfgw64 -q"10^21834-93"
10^21834-93 is composite: RES64: [8EF5236AE00E8EE4] (1.4598s+0.0003s)
C:\pfgw>pfgw64 -a1 -q"10^21834-93"
10^21834-93 is composite: RES64: [0C41339B9D1E7C77] (1.9006s+0.0003s)
C:\pfgw>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>pfgw64 -q"10^114575 – 500001"
10^114575-500001 is composite: RES64: [210AB3A805D8CC8B] (53.3825s+0.0015s)
C:\pfgw>pfgw64 -a1 -q"10^114575-500001"
10^114575-500001 is composite: RES64: [210AB3A805D8CC8B] (63.5933s+0.0026s)
C:\pfgw>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 |
| 10^28607 | – 134861 | + 84549 |
| 10^32370 | – 236397 | + 103309 |
| 10^33875 | – 106329 | + 79813 |
| 10^37638 | – 176109 | + 81867 |
| 10^39143 | – 111399 | + 103779 |
| 10^42906 | – 8549 | + 294093 |
| 10^44411 | – 95669 | + 195391 |
| 10^48174 | – 27311 | + 22611 |
| 10^49679 | – 60063 | + 115581 |
| 10^53442 | – 2007 | + 12621 |
Research from February 1, 2018
When looking closer at these four PFGW failure numbers
I see that the difference between 21834 and 23339 equals that
of between 27102 and 28607 namely 1505. The difference between
23339 and 27102 equals 3763.
Next I wondered if I could extrapolate this alternating pattern
(+ 3763 and + 1505) and see if all emerging numbers fail for PFGW
as well. To my surprise, starting from 18071 up to 53442 they do !
Sloane's OEIS sequences
- A033873, A033874 and A038804
- https://oeis.org/A033873
- https://oeis.org/A033874
- https://oeis.org/A038804
- A003617 and A003618
- https://oeis.org/A003617
- https://oeis.org/A003618
- A096548
- https://oeis.org/A096548
Initials and abbreviations
| AG | = Alexander Gramolin |
| AN | = Anand Nair |
| AR | = Alfred Reich |
| 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 |
| PB | = Paul Bourdelais |
| PDG | = Patrick De Geest |
| PK | = Peter Kaiser |
| RC | = Ray Chandler |
| RP | = Robert Price |
| SB | = Serge Batalov |
| SM | = Stefano Morozzi |
| 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
https://oeis.org/A089675 Numbers n such that 10^n - 3 is a prime. [>407197]
10^n - 7 is always divisible by 3
https://oeis.org/A095714 Numbers n such that 10^n - 9 is a prime. [>221631]
https://oeis.org/A092767 Numbers n such that 10^n - 11 is a prime. [>100000]
10^n - 13 is always divisible by 3
https://oeis.org/A108326 Numbers n such that 10^n - 17 is a prime. [>100000]
10^n - 19 is always divisible by 3
https://oeis.org/A108327 Numbers n such that 10^n - 21 is a prime. [>100000]
https://oeis.org/A108328 Numbers n such that 10^n - 23 is a prime. [>100000]
https://oeis.org/A108329 Numbers n such that 10^n - 27 is a prime. [>100000]
https://oeis.org/A108330 Numbers n such that 10^n - 29 is a prime. [>100000]
10^n - 31 is always divisible by 3
https://oeis.org/A108364 Numbers n such that 10^n - 33 is a prime. [>100000]
10^n - 37 is always divisible by 3
https://oeis.org/A108365 Numbers n such that 10^n - 39 is a prime. [>100000]
https://oeis.org/A178406 Numbers n such that 10^n - 41 is a prime. [>100000]
10^n - 43 is always divisible by 3
https://oeis.org/A178175 Numbers n such that 10^n - 47 is a prime. [>100000]
10^n - 49 is always divisible by 3
https://oeis.org/A178429 Numbers n such that 10^n - 51 is a prime. [>40000]
https://oeis.org/A178430 Numbers n such that 10^n - 53 is a prime. [>100000]
https://oeis.org/A108493 Numbers n such that 10^n - 57 is a prime. [>100000]
https://oeis.org/A108506 Numbers n such that 10^n - 59 is a prime. [>40000]
10^n - 61 is always divisible by 3
https://oeis.org/A178433 Numbers n such that 10^n - 63 is a prime. [>100000]
10^n - 67 is always divisible by 3
https://oeis.org/A177866 Numbers n such that 10^n - 69 is a prime. [>100000]
https://oeis.org/A178434 Numbers n such that 10^n - 71 is a prime. [>100000]
10^n - 73 is always divisible by 3
https://oeis.org/A178436 Numbers n such that 10^n - 77 is a prime. [>100000]
10^n - 79 is always divisible by 3
https://oeis.org/A178437 Numbers n such that 10^n - 81 is a prime. [>100000]
https://oeis.org/A178438 Numbers n such that 10^n - 83 is a prime. [>100000]
https://oeis.org/A108331 Numbers n such that 10^n - 87 is a prime. [>15328]
https://oeis.org/A108332 Numbers n such that 10^n - 89 is a prime. [>40000]
10^n - 91 is always divisible by 3
https://oeis.org/A178531 Numbers n such that 10^n - 93 is a prime. [>100000]
10^n - 97 is always divisible by 3
https://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.
https://oeis.org/A049054 Numbers n such that 10^n + 3 is a prime. [>100000]
https://oeis.org/A088274 Numbers n such that 10^n + 7 is a prime. [>100000]
https://oeis.org/A088275 Numbers n such that 10^n + 9 is a prime. [>100000]
10^n + 11 is always divisible by 3.
https://oeis.org/A095688 Numbers n such that 10^n + 13 is a prime. [>100000]
10^n + 17 is always divisible by 3
https://oeis.org/A108052 Numbers n such that 10^n + 19 is a prime. [>100000]
https://oeis.org/A108050 Numbers n such that 10^n + 21 is a prime. [>100000]
10^n + 23 is always divisible by 3
https://oeis.org/A108312 Numbers n such that 10^n + 27 is a prime. [>100000]
10^n + 29 is always divisible by 3
https://oeis.org/A107083 Numbers n such that 10^n + 31 is a prime. [>100000]
https://oeis.org/A107084 Numbers n such that 10^n + 33 is a prime. [>100000]
https://oeis.org/A135109 Numbers n such that 10^n + 37 is a prime. [>72631]
https://oeis.org/A135108 Numbers n such that 10^n + 39 is a prime. [>81778]
10^n + 41 is always divisible by 3
https://oeis.org/A108049 Numbers n such that 10^n + 43 is a prime. [>35393]
10^n + 47 is always divisible by 3
https://oeis.org/A108054 Numbers n such that 10^n + 49 is a prime. [>80883]
https://oeis.org/A135118 Numbers n such that 10^n + 51 is a prime. [>47026]
10^n + 53 is always divisible by 3
https://oeis.org/A135119 Numbers n such that 10^n + 57 is a prime. [>100000]
10^n + 59 is always divisible by 3
https://oeis.org/A135116 Numbers n such that 10^n + 61 is a prime. [>100000]
https://oeis.org/A135115 Numbers n such that 10^n + 63 is a prime. [>100000]
https://oeis.org/A135113 Numbers n such that 10^n + 67 is a prime. [>100000]
https://oeis.org/A135114 Numbers n such that 10^n + 69 is a prime. [>100000]
10^n + 71 is always divisible by 3
https://oeis.org/A135132 Numbers n such that 10^n + 73 is a prime. [>100000]
10^n + 77 is always divisible by 3
https://oeis.org/A135131 Numbers n such that 10^n + 79 is a prime. [>63916]
https://oeis.org/A137848 Numbers n such that 10^n + 81 is a prime. [>100000]
10^n + 83 is always divisible by 3
https://oeis.org/A135117 Numbers n such that 10^n + 87 is a prime. [>100000]
10^n + 89 is always divisible by 3
https://oeis.org/A110918 Numbers n such that 10^n + 91 is a prime. [>100000]
https://oeis.org/A135112 Numbers n such that 10^n + 93 is a prime. [>100000]
https://oeis.org/A135107 Numbers n such that 10^n + 97 is a prime. [>100000]
https://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/won161.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
Happy prp hunting...
Largest gaps sorted by merit > 10
How to calculate the merit of a gap ?
merit = gapsize / ( x * ln(10) )
| merit | gap | x | by |
|---|---|---|---|
| 12,494 | 856060 | 29758 | PDG |
| 12,207 | 398370 | 14173 | MLB |
| 11,510 | 738970 | 27882 | PDG |
| 11,238 | 417864 | 16148 | AN |
| 11,149 | 286796 | 11172 | PDG |
| 11,138 | 644032 | 25112 | PDG |
| 10,939 | 540038 | 21440 | PDG |
| 10,134 | 450628 | 19311 | PDG |
| 10,133 | 725724 | 31103 | PDG |
| 10,067 | 664320 | 28658 | PDG |
| 10,036 | 430500 | 18630 | PDG |
Record prime gaps
- Between 10^183546 – 584297 and 10^183546 + 1547323 is a prime gap of 2131620 Found on February 4, 2021
- Between 10^148934 – 512057 and 10^148934 + 1530019 is a prime gap of 2042076 Found on January 20, 2021
- Between 10^153935 – 1000001 and 10^153935 + 970801 is a prime gap of 1970802 Found on May 2, 2021
- Between 10^400000 – 339233 and 10^400000 + 1415223 is a prime gap of 1754456 Found on October 28, 2020
- Between 10^90960 – 610659 and 10^90960 + 1072117 is a prime gap of 1682776 Found on June 24, 2019
- Between 10^164216 – 891051 and 10^164216 + 780639 is a prime gap of 1671690 Found on June 25, 2021
- Between 10^173947 – 286391 and 10^173947 + 1372429 is a prime gap of 1658820 Found on Aug 17, 2021
- Between 10^193566 – 454041 and 10^193566 + 1126339 is a prime gap of 1580380 Found on Sep 10, 2021
- Between 10^150000 – 245703 and 10^150000 + 1158603 is a prime gap of 1404306 Found on September 2, 2018
- Between 10^124248 – 501903 and 10^124248 + 899089 is a prime gap of 1400992 Found on March 5, 2021
- Between 10^117142 – 966057 and 10^117142 + 209223 is a prime gap of 1175280 Found on April 15, 2018
- 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 10^29758 – 121593 and 10^29758 + 734467 is a prime gap of 856060
- Between 10^77418 – 33 and 10^77418 + 841683 is a prime gap of 841716
- Between 10^147843 – 182891 and 10^147843 + 620967 is a prime gap of 803858
- 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 !
Record displacement value > 800.000
- 10^183546 + 1547323
- 10^148934 + 1530019
- 10^400000 + 1415223
- 10^173947 + 1372429
- 10^90101 + 1225537
- 10^150000 + 1158603
- 10^193566 + 1126339
- 10^90960 + 1072117
- 10^153935 – 1000001
- 10^117142 – 966057
- 10^153935 + 970801
- 10^91230 – 911141
- 10^124248 + 899089
- 10^164216 + 891051
- 10^77418 + 841683
- 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^(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 :
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 :
Same 'outerbounds'.
Identical displacements for consecutive values x
|
Three consecutives
|
Two consecutives
|
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 
http://groups.yahoo.com/group/primenumbers/message/138 (10^999+7 and Titanix) by MLB
10^3669 ± y displacement –y = 9663 is 'reverse' of exponent 3669.
10^5020 ± y http://groups.yahoo.com/group/primenumbers/message/3031 (Prime Gap of 82794) by MLB
10^5028 ± y http://groups.yahoo.com/group/primenumbers/message/3100 (Prime Gap of 48116) by MLB
10^9999 ± y 10^9999 + 33603 is genuine prime http://www.worldofnumbers.com/won144.htm
10^10000 ± y http://groups.yahoo.com/group/primenumbers/message/17440 ([Note : many errors!] 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 http://groups.yahoo.com/group/primenumbers/message/2706 ([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 http://groups.yahoo.com/group/primenumbers/message/17459 ([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 !
10^99999 + 309403 was announced public via a Yahoo Primeform Message http://groups.yahoo.com/group/primeform/message/4103 by Daniel Heuer
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
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+, 50225+, 2222 1000*, 2879*, 3333 3197+, 4213+, 7042+, 8914-, 15485+, 35195+ 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+, 30782+, 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-, 32093+, 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-, 27798-, 3232 625*, 3434 690*, 1939*, 3636 1246*, 3737 7420-, 27032-, 3838 2289*, 3939 1498+, 12918+, 21435-, 24525-, 26604+, 35560+, 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-, 30379-, 5252 4567*, 5353 3131+, 5656 7072*, 5757 2062-, 3162-, 4478-, 4587+, 8130-, 15560-, 18650+, 20025-, 29436-, 5959 1472+, 1903+, 6934+, 6060 18213*, 6363 793-, 1756+, 2599+, 2966-, 4439+, 6522-, 15045-, 18355-, 31057+, 6767 4298-, 9159-, 17538-, 30712-, 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*, 27919*, 8686 3364*, 3765*, 8787 5668+, 6807+, 43500-, 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*, 30672*, 123123 23511-, 124124 21771*, 161161 17293+, 183183 23989-, 28166-, 186186 20562*, 204204 36999*, 214214 29861*, 221221 28059+, 230230 30694*, 239239 35630+, 264264 15525*, 278278 29857*, 279279 29170+, 284284 30242*, 288288 50240*, 434434 40800*, 660660 88888*,
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
27416 -4263 +85737 *90000 * rounded gap
28359 -10391 +102391 *112782 Compare 10391 with 10(2)391
28596 -171717 +105631 *277348 Nice number 17_17_17
28600 -101181 +1447 *102628 Displacement -101181 equals that of 10^28733
28733 -101181 +5499 *106680 Displacement -101181 equals that of 10^28600
29020 -82719 +867 *83586 PDG (+ 8 maa 2008) < +29833 by mlb
30435 -9263 +91693 *100956 * Birthday of PDG Okt/09/1956
31962 -50657 +9343 *60000 * rounded gap
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
150000 -245703 +1158603 *1404306 First found displacement above one million
200000 -45557 +571791 *617348 Searched from august 2 to 21, 2019
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 !
|
102237 – 10203 1011672 – 10203 1012037 – 10203 1021236 – 10203 1030840 – 10203 |
106100 + 10203 107554 + 10203 107805 + 10203 1026645 + 10203 1030726 + 10203 |
and their rearrangements like 10^21095 + 20301 10^7881 + 30201 10^7687 with gap *30102
List to update regularly
+/- 313 add to palprim4.htm +/- 353 add to palprim4.htm +/- 373 add to palprim4.htm +/- 383 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 +/- 16661 add to palprim3.htm +/- 1777771 add to wonplate 161 (not borderprime per se)
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Patrick De Geest - Belgium
- Short Bio - Some PicturesE-mail address : pdg@worldofnumbers.com