A157764 -id:A157764 - cp's OEIS frontend

A375215 Primes of the form p^16 + 2^16, where p is prime (see also A157764).

15496731425178936435099327796097, 295216374856540727739668685343937, 4579937329576774398276408998557697, 19419444565344683427626434801775297, 643780251284828743866259724717471297, 1110832290554380967776058484990830657, 57196271293373441589892672200988689857, 75456166331666628614079195878996262017

Offset: 1

Mykhailo Papenko, Oct 17 2024

It is conjectured that solutions for p1^n + p2^n = p3 (where p1, p2, and p3 are all primes and n is a natural number) exist only when n is itself a power of two (when n is a number in A000079); and would have infinitely many solutions.

But it's known that either p1 or p2 must be a 2.

			a(1) = 89^16 + 2^16 = 15496731425178936435099327796097, which is prime.
a(2) = 107^16 + 2^16 = 295216374856540727739668685343937, which is prime.
a(3) = 127^16 + 2^16 = 4579937329576774398276408998557697, which is prime.
a(4) = 139^16 + 2^16 = 19419444565344683427626434801775297, which is prime.
a(5) = 173^16 + 2^16 = 643780251284828743866259724717471297, which is prime.
a(6) = 179^16 + 2^16 = 1110832290554380967776058484990830657, which is prime.
a(7) = 229^16 + 2^16 = 57196271293373441589892672200988689857, which is prime.
a(8) = 233^16 + 2^16 = 75456166331666628614079195878996262017, which is prime.
a(9) = 349^16 + 2^16 = 48440300802975619860301347588732379759937, which is prime.
a(10) = 421^16 + 2^16 = 973898133213875918230007677219773667320257, which is prime.

Mykhailo Papenko, Table of n, a(n) for n = 1..29908
Mykhailo Papenko, Primes-Made-Up-of-Primes, Github.

The corresponding primes p are in A157764.

/* see link for code with instructions */

Select[Table[Prime[p]^16+2^16,{p,60}],PrimeQ] (* James C. McMahon, Nov 18 2024 *)

a(n) = A157764(n)^16 + 2^16.

A158361 Primes p with property that Q = p^4 + 2^4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 37, 41, 59, 61, 71, 79, 97, 131, 139, 179, 223, 227, 229, 241, 283, 313, 317, 359, 367, 379, 383, 389, 439, 449, 461, 487, 503, 521, 569, 593, 617, 619, 631, 661, 683, 709, 733, 811, 821, 853, 911, 977, 1049, 1061, 1063, 1069, 1091, 1093, 1117

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2009

Q is always congruent to 1 (mod 4).
Q is divisible by 17 if p is congruent to 1, 4, 13, or 16 (mod 17).
It is conjectured that sequence a(n) is infinite.
Q is in A094479. - Zak Seidov, Jul 08 2020

			3 is in the sequence since for p=3: p^4+2^4 = 3^4+16 = 97 is prime.
29 is not in the sequence since 29^4+2^4 = 707297 = 73 x 9689 is not prime.

Leonard E. Dickson, History of the Theory of numbers, vol. I, Dover Publications 2005
Richard Guy, "Unsolved Problems in Number Theory"

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062324, A094479, A157950, A157764.

[p: p in PrimesUpTo(2000) | IsPrime(p^4+16)]; // Vincenzo Librandi, Jun 18 2014

Select[Range[10^3], PrimeQ[#] && PrimeQ[#^4 + 16] &] (* Vincenzo Librandi, Jun 18 2014 *)
Select[Prime[Range[200]],PrimeQ[#^4+16]&] (* Harvey P. Dale, Jun 23 2014 *)

isA158361(n) = isprime(n) && isprime(n^4+16)

Corrected and edited by Michael B. Porter, Dec 17 2009

A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

Original entry on oeis.org

13, 89, 137, 29, 107, 223, 37, 59, 127, 331, 113, 53, 101, 139, 389, 113, 223, 181, 103, 173, 491, 13, 1223, 5279, 491, 109, 179, 563, 1151, 181, 1277, 7517, 547, 181, 229, 647, 43, 2153, 761, 1993, 8039, 619, 199, 233, 701, 53, 271, 3559, 4133, 2399, 9833, 661, 379, 349, 773

Offset: 3

Jonathan Vos Post, Aug 15 2007

These were computed by Ignacio Larrosa Cañestro, who cautions that some are only probable primes. The k=3 row is A157950. The main diagonal is A132261.

			The array begins:
   n  |   1    2    3    4    5    6     7     8     9    10
  ----+--------------------------------------------------------
  k=3 |  13  137  223  331  389  491   563   647   701   773
  k=4 |  89  107  127  139  173  179   229   233   349   421
  k=5 |  29   59  101  103  109  181   199   379   769   881
  k=6 |  37   53  181  491  547  619   661   677   911   941
  k=7 | 113  223 5279 7517 8039 9833 12197 13757 21467 23447
  k=8 | 113 1223 1277 1993 2399 9349  9739 10211 10973 11059

Cf. A000040, A062324, A132261, A157764, A157950, A158361, A158477.

More terms from Jinyuan Wang, Feb 01 2022

A157950 Primes p such that p^8 + 2^8 is prime.

Original entry on oeis.org

13, 137, 223, 331, 389, 491, 563, 647, 701, 773, 797, 1063, 1181, 1531, 1579, 1811, 2027, 2087, 2269, 2333, 2393, 2617, 2687, 2699, 2857, 3313, 3467, 3623, 3637, 3691, 3739, 3761, 3863, 3877, 4133, 4201, 4283, 4297, 4877, 5023, 5839, 5897, 6043, 6053

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 10 2009

nonn

17 divides p^8 + 2^8 if k is odd and p = 17k +- 6, 17k +- 10, 17k +- 12, 17k +- 14, so only 8 integers p in each interval of length 34 need to be tested for the primality of p and of p^8 + 2^8: those of the forms p = 17k +- 2 (which yield terms 223, 389, 491, 563, 797, 1579, 3313, 3623, 3691, ...), p = 17k +- 4 (which yield terms 13, 701, 2027, 2087, 2333, 2393, 2699, ...), p = 17k +-8 (which yield terms 331, 773, 1063, 1181, 1811, 2269, ...), and p = 17k +-16 (which yield terms 137, 647, 1531, 2617, 2687, 2857, 3467, 3637, ...).

It is conjectured that this sequence is infinite.

			n=11: 11^8 + 2^8 = 214359137 = 17 * 241 * 52321, not prime, so 11 is not a term;
n=13: 13^8 + 2^8 = 815730977 is prime, so 13 is a term.

Leonard E. Dickson, History of the Theory of Numbers.
Richard Guy, Unsolved Problems in Number Theory.

Cf. A062324, A157764.

a := proc (n) if isprime(ithprime(n)^8+256) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 900); # Emeric Deutsch, Mar 14 2009

Definition corrected by Emeric Deutsch, Mar 14 2009
Extended by Emeric Deutsch, Mar 14 2009
Edited by Jon E. Schoenfield, Jan 29 2019

A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

Original entry on oeis.org

29, 59, 101, 103, 109, 181, 199, 379, 769, 881, 919, 977, 1097, 1213, 1303, 1583, 2099, 2113, 2441, 2521, 2617, 2777, 3067, 3739, 4133, 4289, 4519, 4931, 5039, 5113, 5227, 5417, 5743, 5783, 6143, 6373, 6691, 8053, 8209, 8287, 8513, 9109, 9203, 9689, 9787, 9923, 9941

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2009

nonn

1) Q=(p^16)^2+(2^16)^2 only for Q=4k+1 because of Fermat/Euler/Lagrange theorem concerning prime as sum of two squares.
2) It is conjectured that sequence a(n) is infinite.
3) Note the twin prime: a(3)=101, a(4)=103.
The next set of twins is a(101)=30557, a(102)=30559. - Robert Israel, Apr 05 2016

			p=3: 3^32+2^32=1853024483819137 = 1153 x 1607133116929 no prime;
also for following primes p=5, 7, 11, 13, 17, 19, 23: Q(p) no prime;
p=29: 29^32+2^32=62623297589448778360828428329074752313100292737 is prime => a(1)=29.

Richard E. Crandall, Carl Pomerance, Prime Numbers: A Computational Perspective, Springer 2001.
Leonard E. Dickson, History of the Theory of Numbers, Dover Pub. Inc., 2005.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062324, A157764, A157950, A158361.

select(t -> isprime(t) and isprime(t^32 + 2^32), [seq(i,i=3..10000,2)]); # Robert Israel, Apr 05 2016

With[{c=2^32},Select[Prime[Range[1300]],PrimeQ[#^32+c]&]] (* Harvey P. Dale, May 04 2018 *)

isA158477(n) = isprime(n) && isprime(n^32+4294967296) \\ Michael B. Porter, Dec 17 2009

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p^32+2^32), print1(p, ", "))); \\ Altug Alkan, Apr 05 2016

n^32+2^32 and n to be prime.

A378490 Least prime p such that p^(2^n) + 2^(2^n) is prime.

Original entry on oeis.org

3, 3, 3, 13, 89, 29, 37, 113, 113, 13, 1151, 43, 53, 5503

Offset: 0

Jean-Marc Rebert, Nov 28 2024

			a(1) = 3, because 3^(2^1) + 2^(2^1) = 13 is prime and no smaller prime satisfies the condition.

Cf. A001146, A132260, A157764, A375215.

a(n) = my(p=2, q=2^n); while (!ispseudoprime(p^q + 2^q), p=nextprime(p+1)); p; \\ Michel Marcus, Nov 28 2024

a(n) = A132260(n,1) for n>=3.

cp's OEIS Frontend

A375215 Primes of the form p^16 + 2^16, where p is prime (see also A157764).

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A158361 Primes p with property that Q = p^4 + 2^4 is prime.

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A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

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A157950 Primes p such that p^8 + 2^8 is prime.

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A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

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Maple

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A378490 Least prime p such that p^(2^n) + 2^(2^n) is prime.

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PARI

Formula