A345213 Primes p such that q^r == r^q (mod p), where p,q,r are consecutive primes.

2, 11, 29, 59, 71, 149, 269, 431, 569, 599, 727, 1031, 1061, 1229, 1289, 1319, 1451, 1619, 2129, 2339, 2381, 2549, 2711, 2789, 3299, 3539, 4019, 4049, 4091, 4649, 4721, 5099, 5441, 5519, 5639, 5741, 5849, 6269, 6359, 6569, 6701, 6959, 7211, 8009, 8999, 9041, 9341, 10091, 10859, 11489, 11831

Offset: 1

Author

J. M. Bergot and Robert Israel, Jun 10 2021

Keywords

• nonn

Comments

Terms not in A049437 include 2, 11, 727 and 22571. Are there others?

Are there primes p other than 7 such that p^q == q^p (mod r), or primes p other than 41 such that p^r == r^p (mod q), where p,q,r are consecutive primes?

Examples

			a(3) = 29 is a term because 29, 31 and 37 are consecutive primes and 37^31 == 31^37 == 19 (mod 29).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Includes A049437 except for 3.

Programs

Maple

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  if q&^r - r&^q mod p = 0 then count:= count+1; R:= R, p fi
od:
R;