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

%I A136191 #11 May 01 2025 08:05:44
%S A136191 13,43,53,113,193,223,283,563,613,643,743,773,1033,1193,1453,1483,
%T A136191 1543,1583,1663,1733,2143,2393,2503,2843,3163,3413,3433,3793,3823,
%U A136191 4133,4463,4483,4523,4603,4673,4813,5443,5743,5953,6073,6133,6163,6553,6733,6863
%N A136191 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal being 3.
%C A136191 Except for p=5, the decimals in A092110 end in 3 or 7.
%C A136191 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.
%t A136191 Select[Prime[Range[1000]],AllTrue[{2#-3,2#+3},PrimeQ]&&IntegerDigits[#][[-1]]==3&] (* _James C. McMahon_, Apr 30 2025 *)
%o A136191 (PARI) isok(n)  = (n % 10 == 3) && isprime(n) && isprime(2*n-3) && isprime(2*n+3); \\ _Michel Marcus_, Sep 02 2013
%Y A136191 Intersection of A092110 and A017305.
%Y A136191 Cf. A136192.
%K A136191 base,nonn
%O A136191 1,1
%A A136191 _Carlos Alves_, Dec 20 2007