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The threedigit palindromic primes

ID Number: A002385 (Formerly M0670 and N0247)
Sequence:  2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,
           929,10301,10501,10601,11311,11411,12421,12721,12821,13331,13831,13931,
           14341,14741,15451,15551,16061,16361,16561,16661,17471,17971,18181
Name:      Palindromic primes.
References A. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228.
Links:     K. S. Brown, On General Palindromic Numbers
           P. De Geest, World!Of Palindromic Primes
Keywords:  nonn,base,nice,easy
Offset:    1
Author(s): njas,sp

101

131

151

181

191

313

313 as displacement to the powers of ten such that they are also the smallest (probable) primes from that axis 10180+313 10661+313 102160+313 102569+313 106012+313

Shyam Sunder Gupta's entry [ June 7, 2002 ] [email] [site] "313 is the smallest happy number which is a multi-digit palindromic prime."

313 divides the following two primorial constructions 73# – 1 = the product of the first 21 primes (from 2 to 73) – 1 239# + 1 = the product of the 52 first primes (from 2 to 239) + 1 Note that (73 + 239) + 1 = 313 Hisanori Mishima, Factorization results PI Pn - 1 (n = 1 to 110) Hisanori Mishima, Factorization results PI Pn + 1 (n = 1 to 110)

Starting with composite number 39 and applying the procedure Repeated Factorization of Concatenated Primefactors we arrive at 39 = 3 x 13 which is 313 after just one step.

313 is the largest palindromic substring of 322 or 911 Both expansions are equal to 31381059609 (See OEIS A046261 and A046267 )

The palstring 7097907 was found at position 3135373 counting from the first digit after the decimal point of  . [ Pi-Search Puzzle Page ] I am still not sure whether this item belongs to this 3135373 or to the 3135373 section. Some suggest I should use it as the first item for section 3135373... Anyway it is a neat fact that these three consecutive palprimes occur like this !

313 is the sum of the squares of two consecutive numbers 313 = 122 + 132

* 313 is the smallest prime having smallest primitive root 10, i.e. 313 is the smallest prime
  such that 10 as the smallest base to make it a full reptend prime.

* 313 written in base 2 is 100111001, which is also palindromic.

* 313 is the number of intersections when all the diagonals of a regular dodecagon are drawn.

* 313 is the largest known generalized half-Fermat prime in base 5 (of the form (5^(2^n)+1)/2).

* 313 is the smallest length of a prime repunit in base 35 (in all smaller base which is not perfect
  power (where repunits can be factored algebraically), there is a prime repunit with length < 20).

* 313 is the second-largest known prime p such that the Wagstaff number (2^p+1)/3 and
  the Lucas number L(p) are both primes (the largest known such p is 10691).

* 313 is the largest known prime that divides a unitary perfect number.

Catch Me If You Can
[ March 30, 2008 ] G. L. Honaker

Did you ever see the film Catch Me If You Can (2002) ?

'313' - Enoch Haga's key to containment of the Beast !

Enoch Haga, a retired teacher (taught mathematics, computer programming and business classes for over
35 years in California public schools) - more about the author - currently living in Livermore, east of San Francisco,
told me that the '313 stuff' started as an innocent challenge inspired by his friend G. L. Honaker, Jr.

Enoch likes to string this '313 stuff' together with some kind of flamboyant narrative. Now, Sri Mighty Flighty
of Cattapurr, Chief Guru of California (CGC), is a convenient scapegoat for Enoch, someone on whom he can
blame his mistakes. Following is his esoteric story of the containment of the beast 666 using palprime 313 !

"The other day while waiting for some PalPrimes to pop up, I happened to
step outside and inhale some smog. Shortly thereafter, I received a revelation:
I was told that I am controlled by Sri Mighty Flighty of Cattapurr,
Chief Guru of California, and as such the SF has authorized me to issue
proclamations in my name under his authority.

Since Re 13:18 clearly states the number of the wild beast to be 666, and
since this mark is to be placed on the forehead or on the right hand (my
keyboard hand! and mouse controller!), I see that the Wild Beast, being a
powerful fallen angel, is infinitely uncontained. Now the Wild Beast can
only be contained by an unfallen angel, indeed one who is sent out to do
battle. This, it has been revealed to be Michael, whose number is 99. The
SF is content to watch them both do battle forever, and will not intervene
so as to spoil the fun.
Therefore, a Wild Beast, or Beast, is 666, and a Michael is 99. Each of
these angels has legions of angels, b's or beasts for the Beast, and little
a's or angels for Michael.

6 = beastly germ
66 = infectious germ
666 = virulent beast or the Wild Beast or B
99 = fighting angel, slayer of dragons, Michael, or M

Both B and M have angels in two ranks:

Michael's 1^st rank
9 1's x 2 or 18 (111111111666111111111)
3 3's x 2 or 18 (333666333)

Michael's 2^nd rank
2 7's x 2 or 28 (7766677)
2 9's x 2 or 36 (9966699) DANGER! 4 x 9 = 36; this can unleash 6 more
beastly germs and send them out to the four corners of the earth!
Just two 3's do not contain and in fact may be sucked in by the B to gain
energy! (36663 - the 3+3 adds a germ! moreover the total is now 24, and
24 / 4 = a drunken B)
5 2's x 2 = 20 (2222266622222) contains B and gives edge to M
3 4's x 2 = 24 (444666444) contains B BUT power could be leaked to B, as
24/4 = 6, and that 6 could defect from M to B

Beware of 3 2's together as they may create a beastly germ!

9966699 is a double M, highly effective in containment, even better:
the triple M: 999666999, and no angels can defect as 27 isn't divisible by 6!

Beware of situations such as this: 6661666; notice that one of Michael's 1^st
rank angels is vulnerable and may be captured and even eaten!

It has been revealed to me that the B can be contained only through
DIVISION, thus 1318/2 = 659, a PRIME (Recall Re 13:18). Thus if we divide
666/2 we obtain 333, a reduced B. Now, if we replace the middle three with
an angel of the first rank, 1, we obtain 313 (the Number of the Duck), the
KEY TO CONTAINMENT OF THE BEAST.

313666313 does the trick, as if there is an attack of 6 in either direction,
it is immediately overwhelmed by 7 on either side. Nevertheless, the B will
always keep trying!

The vision fades...

Sri Mighty Flighty of Cattapurr, CGC"

Some notes

Donald Duck ?
Visited http://www.brucehamilton.com/gladstone/disneycomics/prevish.html.
Now I understand why Enoch calls 313 the Number of the Duck !

  "313 is the license plate number on Donald Duck's automobile!"

In the constant  

The string 313 found at position 858 counting from the first digit after the decimal point.
Palindrome 78387 follows immediately !
[ See also in OEIS database sequence with index number A038101 ]
The string 666 found at position 2440 counting from the first digit after the decimal point.
Palprime 727 follows immediately !

The Duck Army and 313
G. L. Honaker, Jr.'s motto "QUACK ALL THAT YOU CAN QUACK!" [ February 10, 2000 ]

Interesting sources

Floating around in space for 313 days.

G. L. Honaker, Jr. reports the following fact :

The Russian astronaut Sergei Krikalev returned to Earth after spending
exactly 313 days in space. In space, astronauts (or cosmonauts) can expect their
heartbeat to slow and suffer from space sickness...

Websource : All Systems Go: Bar Code in Space

A duck to the duckth.
Facts from Enoch Haga

There is only one duck in the expansion of 313^313 - a 782 digit number !
After the 406^th digit you'll find the one and only 313.

 1274610970300263200888968784417700338335618419643837688920311024503746596164896
41249810762237495002232897811996972160699514037144492043557032105601277552468661
20534521961427863116198351965979693980604137846614187055474451810271643948594982
78737159667679625114138816298363181256549958711646379057979446259953191257493281
29013646348865942517186824182307729704748276081791721496521780740927481599977133
34136183135645205767361477983753414314622304158485220665043016109249805729529425
53755954502039432994417164077215336872224664149250572051320157903542207932768578
47789998066360064670894400346948871692058732092734190231554127965176337936104389
84279564833034071143284244326458520422902338436492705306298219928193213795513096
016308040297068762469448765341747383352706249717060033734444153

313 divides evenly any string of 312 identical digits 1, 2, 3, 4, 5, 6, 7, 8 or 9.
312 is the lowest value.
There is a right triangle where the hypothenuse is 313 and the sides are 312 and 25.

313 and 11

From “Prime Curios!” by C.K. Caldwell & G.L. Honaker, Jr.
as Wells as from “Curious and Interesting Numbers (Revised edition 1997)”

The Pellian and 313

313 a Binary Gleichniszahlen-Reihe term

313 is a term from the 'Binary Gleichniszahlen-Reihe (BGR)' when converted into decimal !
Consult OEIS sequences A045998, A045999 and A048522.

Describe              after mod 2              into decimal
                      (A045998)                (A048522)
-----------------------------------------------------------------------------
1                     1                        1
11                    11                       3
21                    01                       1
1011                  1011                     11
111021                111001                   57
312011                110011                   51
212021                010001                   17
10113011              10111011                 187
1110311021            1110111001               953
3110312011            1110110011               947
3110212021            1110010001               913
3120113011            1100111011               827
2120311021            0100111001               313 ---> Number of the Duck!
...

Anno Domini 313

In 313 AD Constantine was in the 7^th year of his reign.

To find out what happened on any day, including March 13 (3-13) go to:
Today In History

313 revisited.

Enoch Haga wrote a book “Exploring Prime Numbers on Your PC”,
2nd ed. 1998, ISBN 1-885794-16-9 (3rd ed. out soon) on primes
that also set off G. L. Honaker, Jr. to thinker with the palprime 313.

Beginning on page 5,

"Because 313 is my favorite number and palprime, I use it as an example to try my algorithm for testing numbers for primality.
Of course 313 became my favorite number because each day when I left my teaching job, I happened to notice that the time
was 3:13 pm. Usually I left home at 7:11 am because it is a double-lucky number at the craps tables! I mentioned that 313 is a
palindromic number, and asked “How many palindromic primes are there? It might be fun to try to find out...”

“I'm not going to let you leave this book without learning more about my favorite number, 313. This is a three-digit palindrome
whose digits sum to a lucky number, 3+1+3 =7. . . . Two of these digits, 3 and 7, are also prime, and the other isn't composite!
Twice the lucky number less 1 is an unlucky number: 2*7-1=13. The digits of 14 sum to 5, another prime. The unlucky number,
13, with its digits reversed is a prime, 31, terminating a five-digit palindrome: 1 3 1 3 1. (A prime, such as 13, which is reversible
to another prime, such as 31, is called an emirp.) The first digit of 313, when squared, factors the larger palindrome evenly:
13131/9 = 1459. The digits of this second factor, 1459, which is also a prime, when summed and summed again (1+4+5+9=19,
and 9+1=10), give us the base of our number system, 10. To clinch it all, 313 = 12^2 + 13^2: 144 + 169 = 313, the sum of two
consecutive squares. Can you discover anything else about 313? ”

353

* 353 is the smallest non-single digit palindromic prime whose digits are all primes.

* 353 is the second irregular prime with irregular index > 1 (the first such irregular prime is 157).

* The repunit R353 is the first repunit whose complete factorization is unknown.

* 353 is the smallest positive integer whose 4-th power is the sum of four 4-th powers of smaller
  positive integers (353^4 = 30^4 + 120^4 + 272^4 + 315^4).

* 353 is the smallest odd number n such that there are no known primes or PRPs of the form n^k–2,
  note that for n=353, there is also no known primes or PRPs of the form n^k–(n–1), also the
  smallest prime or PRPs of the form n^k–(n+1) is also large, it is 353^2832–354.

373

373 as displacement to the powers of ten such that they are also the smallest (probable) primes from that axis 1087+373 105947+373 107987+373 109525+373 1011067+373

373 is a circular prime.

Alberto Hernández Narváez, from Monterrey, México (email) constructed (22/07/1999) the following interesting 8 x 8 record matrix 1 3 1 6 3 3 9 3 1 9 3 4 9 1 9 9 3 3 3 9 1 1 3 9 6 3 3 8 9 2 9 9 9 7 3 7 5 4 7 1 7 3 2 7 1 3 4 7 3 1 9 9 6 7 9 3 9 3 9 6 7 7 9 3 The Hernández matrix has exactly 373 distinct embedded primes ! Read the lines or part of them vertically, horizontally or diagonally, and in both direction. For more information visit Carlos Rivera's PP&P Gordon Lee puzzle.

373 is a prime factor of two Reversed Smarandache Concatenated Numbers The first one is Rsm61 or 616059...7654321 The second one is Rsm167 or 167166165...7654321 Furthermore Sm194 is not completely factored. (See Smarandache Concatenated Numbers) To find them all you need to do is factorize a 373-digit composite number !

The sum of five consecutive primes 373 = 67 + 71 + 73 + 79 + 83

The sum of the squares of five consecutive primes 373 = 32 + 52 + 72 + 112 + 132

373 is a sum of positive powers of its digits. [ OEIS A007532 ] 31 + 73 + 33

The string 373 was found at position 5229 counting from the first digit after the decimal point of  . [ Pi-Search Puzzle Page ] No other 3-digit prime is 'first' found beyond this position.

373 is the only 3-digit palindromic prime 'generator' [ G. L. Honaker, Jr. ] of a length-4 prime chain. Go here to the topic about Self Descriptive Primes

A group of 25 elements can be partitioned in exactly 373 palindromic ways. [ OEIS A025065 - by Clark Kimberling ]

Factorial 199 or 199! has exactly 373 digits. Note that 199 is a prime number. [ OEIS A035065 to A035068 ] 373 can be expressed in three ways as prime1 + prime2 + 1 - (Carlos Rivera) 199 + 173 + 1 | 193 + 179 + 1 | 191 + 181 + 1

Starting with composite number 38 and applying the procedure Repeated Factorization of Concatenated Primefactors we arrive at 38 = 2 x 19 and 219 = 3 x 73 which is 373 after two steps.

Most of us possess or read David Wells' book "Dictionary of Curious and Interesting Numbers", Penguin Books Ltd., Ed. 1988. Alas, there is no entry for the number 373 !

We all live in a palindromic 11-year interval [ 1991 - 2002 ] - An interpalindromicum. 1991 written out in English = ONE THOUSAND NINE HUNDRED NINETY ONE Take A=1, B=2, C=3, ... and the summation yields our number 373 - See PP&P Puzzle 33

Here is a beautiful Magic Square filled only with palindromes  282 737 646 Pay attention, folks, as now follows a very difficult question  919 555 191 Can you guess the missing number in the middle  464   ?   828

Water boils at a temperature of 100° Celsius. Water boils at a temperature of 212° Fahrenheit. Water boils at a temperature of ....... 373° Kelvin !

373 is highly decomposable and transformable into other primes. Every prefix is prime 373 - 37 - 3 | OEIS A024770 Every suffix is prime 373 - 73 - 3 | OEIS A033664 Every permutation of its digits is prime 373 - 337 - 733 | OEIS A003459

373 is palindromic in other bases as well. 37310 = 4549 = 5658 = 113114

373 is the average of its two 'neighbour primes' | OEIS A006562. 373 = ( 367 + 379 ) / 2

* 373 is the largest prime whose non-empty substrings are all primes.

* 373 is the smallest palindromic prime which is also a balanced prime.

* 373 is a permutable prime.

* The smallest generalized Wagstaff prime or PRP in base 373
  (i.e. of the form (373^n–1)/(373+1)) is (373^24007+1)/374.

'373' the king amidst the threedigit palprimes

Let me expand with another rather small palindromic prime namely  373 .
Quite uninteresting at first sight, I hear you utter.
But then take another look at the following table were various random aspects of this number are displayed.
You'll never say again a number is uninteresting.
Mike Keith has written a very interesting paper about this uninteresting? topic.

A relation between 373 and 131
from Henk Bakker of The Netherlands (email) dd. [ May 15, 2001 ]

383

383 as displacement to the powers of ten such that they are also the largest (probable) primes from that axis 10218–383 10842–383 103301–383 103504–383 108365–383 1018225–383

383 is a Woodall prime. It is explained by Matt Parker in the video below. Or 383 = 6*26–1

383 is the sum of the first three threedigit palindromic primes 101 + 131 + 151.

* The smallest prime of the form 383*2^n+1 is very large (383*2^6393+1),
  the previous record is 47*2^n+1, which is 47*2^583+1.

* 383 is the largest known palindromic Woodall prime (the only other such prime is 7).

* 383 is the largest prime base such that there are no pseudoprimes < b–1.

* 383 is the largest base b such that there are no primary pretenders (A000790) < b–1.

* The smallest prime of the form 2*383^n+1, if exists, is > 2*383^1000000+1,
  also if b=383, none of 2*b^n+1, 2*b^n–1, b^n+2, (b–1)*b^n–1, b^n–(b–1),
  have small primes (but b^n–2, (b–1)*b^n+1, b^n+(b–1) have, 383^54+382 is prime).

727

* 727 is the smallest prime with period length 121 in base 2, note that 121 is also palindromic
  number, and 727 and 121 together use two 1's, two 2's, and two 7's.

* 727 = 1! + (1+2)! + (1+2+3)!

757

* For the smallest k such that Phi(k,n/(n-1))*(n-1)^eulerphi(k) (where Phi is the cyclotomic polynomial)
  for 2 ⩽ n ⩽ 1000, this k is the largest for n=757 (the corresponding k is 414), note that 414 is also
  a palindromic number. Also, for k ⩽ 1000, only 5 k-values produce primes: 414, 546, 634, 696, 954,
  all are even numbers; if we require k to be prime, then the smallest k is 96487, see A058013.

* 757 is a 'de Polignac' number (not of the form 2^n+k), see A006285.

* 757 is irregular prime (see A000928), as it divides the numerator of the Bernoulli number B(514).

* 757 is the only one generalized repunit prime in base 27 (11127).

* 757 is the smallest prime p such that 1/p has period length 27 (in decimal) = 0.001321003963011889035667107

* 757 is the smallest palindromic prime that remains prime when its leading digit is replaced by any
  nonzero digit that is a power of 2 [Russo]
  (i.e. 157, 257, 457 and 857).

* 757 is the sum of seven consecutive primes (97 + 101 + 103 + 107 + 109 + 113 + 127)
  but not the sum of any number of consecutive composite numbers.

* For k=757, there is no known prime of the form (k-1)*k^n+1 with n>=1, such primes must have n>100000, and
  (if such prime exists) the smallest such prime will be a minimal prime > b (see WONplate 218) in base b=757
  (this problem is extremely hard if the base (b) is large, according to my research, when the base b has
  (b-1)*eulerphi(b) > 100000 (i.e. A062955(b) > 250000), the largest minimal prime > b would be well over
  a googolplex (10^(10^100)), and thus the minimal prime > b problem could not be solved in the entire life
  of the known universe even with exponentional increases in computing power or with quantum computers,
  the largest minimal prime > b is roughly as a double exponential function of A062955(b), and for base b=757,
  A062955(b) is 571536).

787

* 787 is the largest palindromic minimal prime (in decimal) (see WONplate 218).

* 787 is the smallest prime that can be represented as the sum of a prime and its reversal
  in two different ways [Gupta]
  (443 + 344 and 641 + 146).
  Ref. 1 These are also called Luhn Prime Numbers. Ref. 2 Luhn Primes of Order ω

797

919

929

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