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
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[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 *)
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
KEYWORD
nonn
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2010
STATUS
approved