Palindromic Prime
A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having n=1, 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, ... (OEIS A016115; De Geest) and the total number of palindromic primes less than 10, 10^2, 10^3, ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251). Gupta (2009) has computed the numbers of palindromic primes up to 10^(21).
The following table lists palindromic primes in various small bases.
Banks et al. (2004) proved that almost all palindromes (in any base) are composite, with the precise statement being
where P(x) is the number of palindromic primes <=x and N(x) is the number of palindromic numbers <=x.
The sum of the reciprocals of the palindromic primes converges to approx 1.3240 (OEIS A118064) a number sometimes known as Honaker's constant (Rivera), where the value computed using all palindromic primes <=10^(11) is 1.32398... (M. Keith).
The first few palindromic primes formed by taking n digits in the decimal expansion of pi and reflecting about the last digit are 3, 313, 31415926535897932384626433833462648323979853562951413, ... (OEIS A039954; Caldwell). These numbers are prime for n=1, 2, 27, 151, 461, 2056, ... (OEIS A119351), with no others for n<=56755 (E. W. Weisstein, Mar. 21, 2009).
The first few n such that both n and p_n are palindromic (where p_n is the nth prime) are given by 1, 2, 3, 4, 5, 8114118, ... (OEIS A046942; Rivera), corresponding to p_n of 2, 3, 5, 7, 11, 143787341 (OEIS A046941; Rivera).
Palindromic primes of the form
| pp_n(x)=x^n+(x+1)^n |
for n=2 include 5, 181, 313, 3187813, ... (OEIS A050239; De Geest, Rivera), which occur for x=1, 9, 12, 1262, ... (OEIS A050236; De Geest, Rivera), with no others for n<10^(20) and x<2×10^(10) (De Geest).
The 20 largest known palindromic primes listed by PrimePages are summarized in the following table.
See also
Belphegor Prime, Palindromic Number, Prime NumberExplore with Wolfram|Alpha
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References
Banks, W. D.; Hart, D. N.; and Sakata, M. "Almost All Palindromes Are Composite." Math. Res. Lett. 11, 853-868, 2004.Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical Entertains. New York: Dover, 1964.Caldwell, C. "The Top Twenty: Palindrome." https://t5k.org/top20/page.php?id=53#records.Caldwell, C. "Prime Curios!: 31415...51413 (53-digits)." https://t5k.org/curios/page.php?curio_id=725.De Geest, P. "Palindromic Prime Page 3." https://www.worldofnumbers.com/palprim3.htm.De Geest, P. "Palindromic Sums of Squares of Consecutive Integers." https://www.worldofnumbers.com/sumsquare.htm.De Geest, P. "Palindromic Numbers and Other Recreational Topics." https://www.worldofnumbers.com/index.shtml.De Geest, P. "Palindromic Prime Statistics--The Table." https://www.worldofnumbers.com/palprim1.htm.Gupta, S. S. "Palindromic Primes Up to 10^(21)." 13 Mar 2009. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0903&L=nmbrthry&T=0&F=&S=&P=2104.Jobling, P. "Re: Record Palindrome." 27 Dec 2005. https://www.worldofnumbers.com/YPFM6764.htm.Rivera, C. "Problems & Puzzles: Puzzle 014-Pal-Primes and Sum of Powers." https://www.primepuzzles.net/puzzles/puzz_014.htm.Rivera, C. "Problems & Puzzles: Puzzle 051-Pi Such that Pi is Palprime & i = Palindrome." https://www.primepuzzles.net/puzzles/puzz_051.htm.Rivera, C. "Problems & Puzzles: Puzzle 056-The Honaker's Constant." https://www.primepuzzles.net/puzzles/puzz_056.htm.Sloane, N. J. A. Sequences A002385/M0670, A006341, A016115, A039954, A046941, A046942, A050236, A050239, A050251, A117697, A117698, A117699, A117700, A117701, A117702, A117703, A118064, and A119351 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Palindromic PrimeCite this as:
Weisstein, Eric W. "Palindromic Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PalindromicPrime.html