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%I #7 Sep 26 2013 06:12:43
%S 7,17,67,97,127,137,157,167,487,547,617,647,937,1187,1277,1427,1627,
%T 1847,2027,2297,2437,2467,2477,2617,2857,2927,3137,3457,3727,4007,
%U 4057,4157,5167,5417,5657,6247,6257,7027,7477,7867,8467,8737,8747,9127,9227
%N Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal of p being 7.
%C Except for p=5, the decimals in A092110 end in 3 or 7.
%C 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.
%H Vincenzo Librandi, <a href="/A136192/b136192.txt">Table of n, a(n) for n = 1..1000</a>
%t bpQ[n_]:=Last[IntegerDigits[n]]==7&&And@@PrimeQ[2n+{3,-3}]; Select[Prime[ Range[1200]],bpQ] (* _Harvey P. Dale_, Sep 25 2013 *)
%Y Cf. A092110, A136191.
%K nonn,base
%O 1,1
%A _Carlos Alves_, Dec 20 2007
%E Definition clarified by _Harvey P. Dale_, Sep 25 2013
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