A176371 Primes p such that reversal(p) - 13 is a square.

31, 41, 71, 83, 281, 311, 431, 479, 733, 751, 797, 2011, 2857, 3163, 4373, 4397, 4943, 7541, 7577, 7583, 9413, 9491, 20533, 20731, 20771, 24151, 24547, 24767, 26249, 28979, 31121, 41201, 41609, 43321, 43391, 43753, 45641, 49459, 49463, 49811, 49891

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 16 2010

Keywords

• base
• nonn

Comments

R(n) denotes the Reversal of a natural number n

List of all (p,N) for p < 10^6 - 1:

(*) for emirp pair (p,R(p)), (+) if square base N is a prime

(41,1), (71,2) (+) (*), (83,5) (+), (281,13) (+), (311,10) (*), (431,11) (+), (479,31) (+), (733,18) (*), (751,12) (*), (797,28),

(2011,33), (2857,87), (4373,61) (+), (4397,89) (+), (4943,59) (+), (7541,38), (7577,88) (*), (7583,62), (9413,56), (9491,44) (*), (20533,183), (20731,117), (20771,133), (24151,123), (24547,273), (24767,277) (+), (26249,307) (+), (28979,313) (+), (31121,110) (*), (41201,101) (+),

(41609,301), (43321,111), (43391,139) (+), (43753,189), (45641,121), (49459,309), (49463,191) (+), (49811,109), (49891,141), (71293,198) (*),

(73133,182), (73471,132), (73597,282) (*), (75521,112), (77611,108) (*), (77849,308), (77863,192) (*), (79613,178), (79841,122) (*), (83207,265),

(83231,115), (83243,185), (83299,315), (90031,114) (*), (92801,104), (96431,116) (*), (98057,274)

Examples

			41 = prime(13), R(41) - 13 = 14 - 13 = 1^2, is a term.
71 = prime(20), 17 - 13 = 2^2, is a term.
83 = prime(23), 38 - 13 = 5^2, is a term.
797 = prime(139) = palindromic prime(18), N = 28^2, is also a term.
Note successive terms that are also consecutive primes: p(17) = 7577, p(18) = 7583, p(36) = 49459, p(37) = 49463, p(46) = 77849, p(47) = 77863.
		

References

• W. W. R. Ball, H. S. M.Coxeter: Mathematical Recreations and Essays, Dover Publications, 13th edition, 1987
• O. Fritsche, R. Mischak and T. Krome: Verflixt und zugeknobelt, Mehr mathematische Raetselgeschichten, Rowohlt TB. Nr.62190, 2007
• C. W. Trigg, Primes with Reverses That Are Powers, J. Rec. Math. 17, 1985

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..7605

Crossrefs

Cf. A000040, A000290, A002385, A006567, A007488.

Programs

PARI

isok(n) = {if (! isprime(n), return (0)); d = digits(n); revn = sum(i=1, #d, d[i]*10^(i - 1)); issquare(revn-13);} \\ Michel Marcus, Aug 25 2013

Python

from sympy import isprime
A176371_list, i, j = [], 0, 13
while j < 10**10:
    p = int(str(j)[::-1])
    if j % 10 and isprime(p):
        A176371_list.append(p)
    j += 2*i+1
    i += 1
A176371_list = sorted(A176371_list) # Chai Wah Wu, Dec 17 2015

Extensions

Two more terms 31 and 3163 added by Michel Marcus, Aug 25 2013