OFFSET
1,1
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.
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.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 17 2015
STATUS
approved