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%I #20 May 12 2022 15:06:43
%S 71,191,6551,9767,18119,21647,27527,35447,46271,79631,103391,103967,
%T 121367,127679,161639,207671,241559,254927,264959,273311,380327,
%U 421079,450599,479879,592367,700127,745751,949607,986567,1011599,1013399
%N Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square.
%C 2p + 3(p + 2) = 5p + 6.
%C There are two parametric solutions for natural numbers:
%C (a) p = 5t^2 + 2t - 1, k = 5t + 1, necessarily for a prime p: t = 2s => p = 20s^2 + 4s - 1, k = 10s + 1.
%C If s = 3k + 2 => p of (a) is not prime but a multiple of 3.
%C If the least significant digit of k is 1, solution of (a) for s = (k - 1)/10).
%C (b) p = 5t^2 + 8t + 2, k = 5t + 4, necessarily for a prime p: t = 2s - 1 => p = 20s^2 - 4s - 1, N = 10s-1.
%C If s = 3k + 1 => p of (b) is not prime but a multiple of 3.
%C If the least significant digit of k is 9, solution of (b) for s = (k + 1)/10).
%D Leonard E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
%D Richard K. Guy, Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994.
%D Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.
%H Amiram Eldar, <a href="/A174370/b174370.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, <a href="http://name.umdl.umich.edu/ABV2766.0001.001">vol. 1</a> and <a href="http://name.umdl.umich.edu/ABV2766.0002.001">vol. 2</a>, Leipzig, Berlin, B. G. Teubner, 1909.
%e 71 and 73 are twin primes, 2 * 71 + 3 * 73 = 19^2.
%e 191 and 193 are twin primes, 2 * 191 + 3 * 193 = 31^2.
%t Select[Prime[Range[10^5]], PrimeQ[# + 2] && IntegerQ[Sqrt[2# + 3(# + 2)]] &] (* _Alonso del Arte_, Dec 05 2011 *)
%t Select[(Range[2251]^2 - 6)/5, And @@ PrimeQ[# + {0, 2}] &] (* _Amiram Eldar_, Dec 24 2019 *)
%t Select[Partition[Prime[Range[80000]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[Sqrt[ 2#[[1]]+ 3#[[2]]]]&][[All,1]] (* _Harvey P. Dale_, May 12 2022 *)
%o (PARI) forstep(n=1,1e4,[10,8,10,2],if(isprime(p=n^2\5-1)&&isprime(p+2),print1(p", "))) \\ _Charles R Greathouse IV_, Dec 05 2011
%Y Cf. A001359, A061308, A069496, A119859, A172271, A172494, A173255.
%K nonn
%O 1,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2010
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