A038703 Primes p such that p^2 mod q is odd, where q is the previous prime.

3, 5, 17, 29, 37, 127

Offset: 1

Author

Neil Fernandez, May 01 2000

Keywords

• nonn
• more

Comments

The next term if it exists is > 32452843 = 2000000th prime. Can someone prove this sequence is complete? - Olivier Gérard, Jun 26 2001

To prove that 127 is the last prime, we need to show that prime gaps satisfy prime(k)-prime(k-1)31. Although it is easy to verify this inequality for all known prime gaps, there is no proof for all gaps. - T. D. Noe, Jul 25 2006

Examples

			The first prime with a prime lower than itself is 3. This squared is 9, which when divided by the previous prime 2 leaves remainder 1, which is odd. So 3 is in the sequence. 11 is not in the sequence because 11^2, when divided by the previous prime 7, leaves a remainder of 121 (mod 7) = 2, which is even.
		

Links

• Eric Weisstein's World of Mathematics, Prime Gaps

Crossrefs

Cf. A038702.

Cf. A058188 (number of primes between prime(n) and prime(n)+sqrt(prime(n))).

Programs

Mathematica

Prime /@ Select[ Range[ 2, 100 ], OddQ[ Mod[ Prime[ # ]^2, Prime[ # - 1 ] ] ] & ]
Transpose[Select[Partition[Prime[Range[50]],2,1],OddQ[PowerMod[Last[#],2, First[#]]]&]] [[2]]  (* Harvey P. Dale, May 31 2012 *)

PARI

isok(p) = isprime(p) && (p>2) && (lift(Mod(p, precprime(p-1))^2) % 2); \\ Michel Marcus, Mar 05 2023

Formula

Prime(k) is in the sequence if prime(k)^2 (mod prime(k-1)) is odd.

Extensions

More terms from Olivier Gérard, Jun 26 2001