A057732 Numbers k such that 2^k + 3 is prime.

1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947, 1875552, 1940812, 2205444

Offset: 1

Author

G. L. Honaker, Jr., Oct 29 2000

Keywords

• nice
• nonn

Comments

Some of the larger entries may only correspond to probable primes.

A number k is in this sequence iff A062709(k) is in A057733; this is the case iff A257273(k) is in A125246. - M. F. Hasler, Apr 27 2015

Examples

			For k = 6, 2^6 + 3 = 67 is prime.
For k = 28, 2^28 + 3 = 268435459 is prime.
		

References

• Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001

Links

• Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
• Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+3, PRP Top Records.

Crossrefs

Cf. A050414, A057733, A062709, A094076, A125246, A257273.

Cf. A019434 (primes 2^k+1), this sequence (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

Programs

Magma

[n: n in [0..1000] | IsPrime(2^n+3)]; // Vincenzo Librandi, Apr 27 2015

Mathematica

Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)

PARI

for(n=1, 2200, if(isprime(2^n+3), print1(n, ", ")));

PARI

for (n=1, 2, if (isprime(2^n+3), print1(n, ", "))); for(n=3, 100000, N=2^n+3 ; S=(N-5)/2 ; x=S ; for(j=1, n-1, x=Mod(x^2-2, N)) ; if(x==S , print1(n, ", "))) \\ produces terms corresponding to probable primes, see formula; Tony Reix, Aug 27 2015

Formula

Here is an LLT-like algorithm, using a cycle of the digraph x^2-2 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^k+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(k-1) == S0 then N is prime. - Tony Reix, Aug 27 2015

Extensions

More terms from Jason Earls, Jul 18 2001 and Mike Oakes, Jul 28 2001

a(47)-a(50) from Donovan Johnson 2006, verified by Paul Bourdelais, Mar 22 2012

a(51) is a probable prime based on trial factoring to 1E9 and PRP testing base 3,5,7 (PFGW v3.3.1). Discovered by Paul Bourdelais, Apr 09 2012

a(52)-a(54) from Paul Bourdelais, Jun 18 2019

a(55) from Paul Bourdelais, Jul 16 2019

a(56) from Paul Bourdelais, Apr 22 2020

a(57) from Paul Bourdelais, Jun 12 2020

a(58) from Paul Bourdelais, Aug 04 2020