A340431 Primes p such that, with q the next prime after p, q > p+2 and q^p == q (mod p+q) and p^q == p (mod p+q).

13, 211, 421, 523, 154321, 221941, 1556641, 2377201, 3918757, 4359961, 7842511, 9163873, 20446561, 1501102081, 7578849037, 15724210681, 25522638481, 52966796353, 68999668237, 109926997057, 112417709113, 209826685297, 694503347201, 963374692897

Offset

1,1

Comments

For twin primes p,q we always have p^q == p (mod p+q) and q^p == q (mod p+q).

Links

Table of n, a(n) for n=1..24.

Example

a(3) = 421 is a term because the next prime is 431, 421^431 == 421 (mod 852) and 431^421 == 431 (mod 852).

Maple

q:= 2: R:= NULL:
while p < 10^7 do
  p:= q; q:= nextprime(p);
  if q-p = 2 then next fi;
  if q &^ p mod (p+q) = q and p &^ q mod (p+q) = p then
    R:= R, p;
  fi;
od:
R;

Prog

(Python)
from sympy import nextprime
A340431_list , p = [], 2
while p <= 10**10:
    q = nextprime(p)
    if q > p+2:
        pq = p+q
        if pow(q, p, pq) == q and pow(p, q, pq) == p:
            A340431_list.append(p)
    p = q # Chai Wah Wu, Jan 12 2021
(PARI) upto(n) = my(p=2); forprime(q = nextprime(p+1), n, if(q-p > 2, if(Mod(p, p+q)^q == p, if(Mod(q, p+q)^p == q, print1(p, ", ")))); p = q); \\ Daniel Suteu, Jan 12 2021

Crossrefs

Sequence in context: A296671 A196328 A362550 * A251093 A132542 A069989

Adjacent sequences: A340428 A340429 A340430 * A340432 A340433 A340434

Keyword

nonn,more

Author

J. M. Bergot and Robert Israel, Jan 12 2021

Extensions

a(15)-a(17) from Daniel Suteu, Jan 12 2021

a(18)-a(22) from Chai Wah Wu, Jan 15 2021

a(23)-a(24) from Martin Ehrenstein, Jan 19 2021

Status

approved