A246488 Gridgeman pairs in increasing order: pairs of palindromic primes which differ only in their middle digits whose difference is equal to 1.

2, 3, 181, 191, 373, 383, 787, 797, 919, 929, 10501, 10601, 11311, 11411, 12721, 12821, 13831, 13931, 15451, 15551, 16561, 16661, 19891, 19991, 30103, 30203, 30703, 30803, 32323, 32423, 35053, 35153, 38083, 38183, 70507, 70607, 77377, 77477, 78787, 78887, 93139

Offset

1,1

Comments

It has been conjectured by Norman T. Gridgeman that infinitely many pairs of such primes exist (see second ref.)

Links

Chai Wah Wu, Table of n, a(n) for n = 1..1000 n = 1..100 from Paolo P. Lava.

Mauro Fiorentini, Coppie di Gridgeman (in Italian)

Mauro Fiorentini, Gridgeman (congettura di) (in Italian)

Prime Curios!, 181

Example

181 and 191 is a Gridgeman pair because the two numbers are palindromic primes which differ only in their middle digits. Furthermore their middle digits differ only in one unit: 8 and 8 + 1 = 9.
The same for 30103 and 30203: middle digits are 1 and 1 + 1 = 2.

Maple

T:=proc(n) local i, x; x:=convert(n, base, 10);
add(x[-i]*10^(i-1), i=1..nops(x)) end:
P:=proc(q) local a, b, j, k, n; j:=[]; a:=2; for n from 1 to q do
if (length(a) mod 2)=1 and T(a)=a then
b:=(trunc(a/10^trunc(length(a)/2))); if b mod 10<9 then b:=b+1:
b:=b*10^trunc(length(a)/2)+(a mod 10^trunc(length(a)/2));
if isprime(b) then j:=[op(j), a, b]: fi: fi: fi:
a:=nextprime(a): od; op(j); end: P(10^4);

Mathematica

Select[Partition[Select[Prime[Range[10000]], PalindromeQ], 2, 1], IntegerQ[ Log10[ #[[2]]-#[[1]]]]&]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 02 2020 *)

Prog

(PARI) ispal(v) = {for(i=1, #v\2, if (v[i] != v[#v-i+1], return(0)); ); return(1); };
isgpal(p) = {d = digits(p); if ((#d % 2) && ispal(d) && (ic = #d\2 +1) && (d[ic]<9) && (d[ic]++) && isprime(q=subst(Pol(d), x, 10)), q); }
lista(nn) = {forprime(p=2, nn, if (q=isgpal(p), print1(p, ", ", q, ", ")); ); } \\ Michel Marcus, Aug 29 2014
(Python)
from sympy import isprime
A246488 = [2, 3]
for n in range(1, 10**4):
....s1 = str(n)
....s2 = s1[::-1]
....for m in range(10-1):
........p1 = int(s1+str(m)+s2)
........p2 = int(s1+str(m+1)+s2)
........if isprime(p1) and isprime(p2):
............A246488.append(p1), A246488.append(p2)
.
A246488 = sorted(set(A246488)) # Chai Wah Wu, Sep 05 2014

Crossrefs

Cf. A002385.

Sequence in context: A328257 A042701 A371232 * A106715 A106817 A233318

Adjacent sequences: A246485 A246486 A246487 * A246489 A246490 A246491

Keyword

nonn,base

Author

Paolo P. Lava, Aug 28 2014

Status

approved