A174370 Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square.

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

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2010

Keywords

• nonn

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).

Examples

			71 and 73 are twin primes, 2 * 71 + 3 * 73 = 19^2.
191 and 193 are twin primes, 2 * 191 + 3 * 193 = 31^2.
		

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.

Crossrefs

Cf. A001359, A061308, A069496, A119859, A172271, A172494, A173255.

Programs

Mathematica

Select[Prime[Range[10^5]], PrimeQ[# + 2] && IntegerQ[Sqrt[2# + 3(# + 2)]] &] (* Alonso del Arte, Dec 05 2011 *)
Select[(Range[2251]^2 - 6)/5, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 24 2019 *)
Select[Partition[Prime[Range[80000]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[Sqrt[ 2#[[1]]+ 3#[[2]]]]&][[All,1]] (* Harvey P. Dale, May 12 2022 *)

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