A174734 Prime numbers n such that 2n-1 and 3n-2 are prime.

3, 7, 37, 211, 271, 307, 331, 337, 601, 727, 1171, 1237, 1297, 1531, 1657, 2221, 2281, 2557, 3037, 3061, 3067, 4261, 4447, 4801, 4951, 5227, 5581, 5851, 6151, 6361, 6691, 6841, 6967, 7621, 7681, 7687, 7867, 8017, 8167, 8191, 8287, 8521, 8527, 8647, 8941

Offset: 1

Author

Michel Lagneau, Mar 28 2010

Keywords

• nonn

Comments

If n, 2n-1 and 3n-2 are prime numbers, and if n >= 5, then n*(2*n-1)*(3*n-2) is a Carmichael number (A033502).

Proof: there exist numbers m such that n=6m+1 is prime (if n=6m+5, then 2n-1 = 12m+9 is composite). Let p=(6m+1)(12m+1)(18m+1) = a*b*c. Then p-1 = 6*12*18*m^3 + (6*12 + 6*18 + 12*18)*m^2 + (6 + 12 + 19)*m, so p-1 is divisible by a-1=6m, by b-1=12m, and by c-1=18m; thus p is a Carmichael number.

Examples

			For n=3, 2n-1 = 5, 3n-2 = 7.
For n=7, 2n-1 = 13, 3n-2 = 19 and 7*13*19 = 1729 (a Carmichael number).
For n=37, 2n-1 = 73, 3n-2 = 109 and 37*73*109 = 294409 (a Carmichael number).
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A13.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• W. R. Alford, Andrew Granville, and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
• Richard Pinch, Carmichael numbers up to 10^18, April 2006.
• Richard Pinch, Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.

Crossrefs

Cf. A002476, A002997, A033502.

Programs

Magma

[ n: n in PrimesUpTo(10000) | IsPrime(2*n-1) and IsPrime(3*n-2) ];

Maple

with(numtheory): for n from 2 to 15000 do: if type(n,prime)=true and type(2*n-1,prime)=true and type(3*n-2,prime)=true then print (n):else fi:od:

Mathematica

Select[Prime[Range[1000]], PrimeQ[2*#-1] && PrimeQ[3*#-2]&] (* Vladimir Joseph Stephan Orlovsky, Jan 13 2011 *)

PARI

forprime(p=3,10^3, isprime(2*p-1) && isprime(3*p-2) && print1(p,", ")); \\ Joerg Arndt, Nov 29 2014

Extensions

Typo in term corrected by D. S. McNeil, Nov 20 2010