A261395 Primes p such that (prime(p)-1)^2 = (prime(q)-1)*(prime(r)-1) for some pair of distinct primes q and r.

13, 47, 137, 191, 193, 223, 227, 313, 701, 857, 907, 947, 991, 1009, 1069, 1291, 1531, 1889, 2281, 2411, 2447, 2647, 3181, 3389, 3539, 3593, 4093, 4099, 4409, 4481, 4603, 4721, 5557, 5647, 6581, 6793, 6869, 6961, 7211, 7349, 7523, 7723, 7753, 8461, 8537, 8543, 8807, 9137, 9241, 9281

Offset: 1

Author

Zhi-Wei Sun, Aug 17 2015

Keywords

• nonn

Comments

Conjecture: Let d be any nonzero integer. Then there are infinitely many prime triples (p,q,r) with p,q,r distinct such that (prime(p)+d)^2 = (prime(q)+d)*(prime(r)+d). In other words, the set {prime(p)+d: p is prime} contains infinitely many nontrivial three-term geometric progressions.

Examples

			a(1) = 13 since (prime(13)-1)^2 = (41-1)^2 = 1600 = (5-1)*(401-1) = (prime(3)-1)*(prime(79)-1) with 13, 3, 79 all prime.
a(2) = 47 since (prime(47)-1)^2 = 210^2 = 44100 = 30*1470 = (prime(11)-1)*(prime(233)-1) with 47, 11, 233 all prime.
		

References

• Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..600
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A260082, A261352, A261385.

Programs

Mathematica

Dv[n_]:=Divisors[n]
L[n_]:=Length[Dv[n]]
f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[Do[If[PQ[Part[Dv[f[k]^2],i]+1]&&PQ[Part[Dv[f[k]^2],L[f[k]^2]-i+1]+1],n=n+1;Print[n," ",Prime[k]];Goto[aa]];Continue,{i,1,(L[f[k]^2]-1)/2}];
Label[aa];Continue,{k,1,1150}]