A136191 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal being 3.

13, 43, 53, 113, 193, 223, 283, 563, 613, 643, 743, 773, 1033, 1193, 1453, 1483, 1543, 1583, 1663, 1733, 2143, 2393, 2503, 2843, 3163, 3413, 3433, 3793, 3823, 4133, 4463, 4483, 4523, 4603, 4673, 4813, 5443, 5743, 5953, 6073, 6133, 6163, 6553, 6733, 6863

Offset: 1

Author

Carlos Alves, Dec 20 2007

Keywords

• base
• nonn

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.

Crossrefs

Intersection of A092110 and A017305.

Cf. A136192.

Programs

Mathematica

Select[Prime[Range[1000]],AllTrue[{2#-3,2#+3},PrimeQ]&&IntegerDigits[#][[-1]]==3&] (* James C. McMahon, Apr 30 2025 *)

PARI

isok(n)  = (n % 10 == 3) && isprime(n) && isprime(2*n-3) && isprime(2*n+3); \\ Michel Marcus, Sep 02 2013