login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A174370 Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square. 4
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

Table of n, a(n) for n=1..31.

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 *)

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 21 04:40 EDT 2019. Contains 325189 sequences. (Running on oeis4.)