A136192 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal of p being 7.

7, 17, 67, 97, 127, 137, 157, 167, 487, 547, 617, 647, 937, 1187, 1277, 1427, 1627, 1847, 2027, 2297, 2437, 2467, 2477, 2617, 2857, 2927, 3137, 3457, 3727, 4007, 4057, 4157, 5167, 5417, 5657, 6247, 6257, 7027, 7477, 7867, 8467, 8737, 8747, 9127, 9227

Offset: 1

Author

Carlos Alves, Dec 20 2007

Keywords

• nonn
• base

Comments

Except for p=5, the decimals in A092110 end in 3 or 7.

Theorem: If in the triple (2n-3,n,2n+3) all numbers are primes then n=5 or the decimal representation of n ends in 3 or 7. Proof: Consider Q=(2n-3)n(2n+3), by hypothesis factorized into primes. If n is prime, n=10k+r with r=1,3,7 or 9. We want to exclude r=1 and r=9. Case n=10k+1. Then Q=5(-1+6k+240k^2+800k^3) and 5 is a factor; thus 2n-3=5 or n=5 or 2n+1=5 : this means n=4 (not prime); or n=5 (included); or n=2 (impossible, because 2n-3=1). Case n=10k+9. Then Q=5(567+1926k+2160k^2+800k^3) and 5 is a factor; the arguments, for the previous case, also hold.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A092110, A136191.

Programs

Mathematica

bpQ[n_]:=Last[IntegerDigits[n]]==7&&And@@PrimeQ[2n+{3,-3}]; Select[Prime[ Range[1200]],bpQ] (* Harvey P. Dale, Sep 25 2013 *)

Extensions

Definition clarified by Harvey P. Dale, Sep 25 2013