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 A174370 Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square. 5
 71, 191, 6551, 9767, 18119, 21647, 27527, 35447, 46271, 79631, 103391, 103967, 121367, 127679, 161639, 207671, 241559, 254927, 264959, 273311, 380327, 421079, 450599, 479879, 592367, 700127, 745751, 949607, 986567, 1011599, 1013399 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 2p + 3(p + 2) = 5p + 6. There are two parametric solutions for natural numbers: (a) p = 5t^2 + 2t - 1, k = 5t + 1, necessarily for a prime p: t = 2s => p = 20s^2 + 4s - 1, k = 10s + 1. If s = 3k + 2 => p of (a) is not prime but a multiple of 3. If the least significant digit of k is 1, solution of (a) for s = (k - 1)/10). (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. If s = 3k + 1 => p of (b) is not prime but a multiple of 3. If the least significant digit of k is 9, solution of (b) for s = (k + 1)/10). REFERENCES Leonard E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005. Richard K. Guy, Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994. Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. EXAMPLE 71 and 73 are twin primes, 2 * 71 + 3 * 73 = 19^2. 191 and 193 are twin primes, 2 * 191 + 3 * 193 = 31^2. MATHEMATICA Select[Prime[Range[10^5]], PrimeQ[# + 2] && IntegerQ[Sqrt[2# + 3(# + 2)]] &] (* Alonso del Arte, Dec 05 2011 *) Select[(Range^2 - 6)/5, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 24 2019 *) PROG (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 CROSSREFS Cf. A001359, A061308, A069496, A119859, A172271, A172494, A173255. Sequence in context: A139991 A140007 A023107 * A174454 A142808 A101110 Adjacent sequences:  A174367 A174368 A174369 * A174371 A174372 A174373 KEYWORD nonn AUTHOR Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2010 STATUS approved

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Last modified January 17 16:43 EST 2021. Contains 340247 sequences. (Running on oeis4.)