A028916 -id:A028916 - cp's OEIS frontend

A281792 Primes of the form x^2 + p^4 where x > 0 and p is prime.

17, 41, 97, 137, 181, 241, 277, 337, 457, 641, 661, 757, 769, 821, 857, 881, 977, 1109, 1201, 1237, 1301, 1409, 1697, 2017, 2069, 2389, 2417, 2437, 2617, 2657, 2741, 2801, 3041, 3217, 3301, 3329, 3541, 3557, 3697, 3761, 3989, 4001, 4177, 4241, 4337, 4517, 4721, 5557, 5641, 5857, 6101, 6257, 6481, 6577

Offset: 1

Robert Israel, Jan 30 2017

nonn

Heath-Brown and Li prove an asymptotic formula for the number of terms <= x, in particular showing that the sequence is infinite.

			17 = 1^2 + 2^4
41 = 5^2 + 2^4
97 = 9^2 + 2^4
137 = 11^2 + 2^4
181 = 10^2 + 3^4

Robert Israel, Table of n, a(n) for n = 1..10000
D.R. Heath-Brown and X. Li, Prime values of a^2+p^4, arXiv:1504.00531 [math.NT], 2015; Inventiones Mathematicae page 1-59 (Sep 30 2016), doi:10.1007/s00222-016-0694-0

Subsequence of A028916.

Cf. A199401, A225119.

N:= 10000: # to get all terms <= N
A:= select(isprime, {seq(seq(x^2+y^4, x=1..floor(sqrt(N-y^4))),
y=select(isprime, [$1..floor(N^(1/4))]))}):
sort(convert(A,list)); # Robert Israel, Jan 30 2017

nn = 10000;
Select[Table[x^2+y^4, {y, Select[Range[nn^(1/4)], PrimeQ]}, {x, Sqrt[nn-y^4 ]}] // Flatten, PrimeQ] // Union (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)

list(lim)=if(lim<17, return([])); my(v=List(),p4,t); forstep(a=1,sqrtint(-16+lim\=1),2, if(isprime(t=a^2+16), listput(v,t))); forprime(p=3,sqrtnint(lim-4,4), p4=p^4; forstep(a=2,sqrtint(lim-p4),2, if(isprime(t=p4+a^2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 13 2017

Heath-Brown and Li prove that there are c*x^(3/4)/log^2 x terms up to x, where c = 4*nu*J = 4.79946121442200811438003177..., nu = A199401, and J = A225119. - Charles R Greathouse IV, Aug 21 2017

A291206 Semi-octavan primes: primes of the form x^4 + y^8.

Original entry on oeis.org

2, 17, 257, 337, 881, 1297, 2657, 6577, 10657, 14897, 16561, 28817, 65537, 65617, 66161, 80177, 83777, 149057, 160001, 166561, 260017, 280097, 331777, 391921, 394721, 411361, 463537, 596977, 614657, 621217, 847601, 1055137, 1336337, 1342897, 1682017, 1763137

Offset: 1

Charles R Greathouse IV, Aug 21 2017

nonn

			a(1) = 1^4 + 1^8 = 2.
a(2) = 2^4 + 1^8 = 17.
a(3) = 1^4 + 2^8 = 257.
a(4) = 3^4 + 2^8 = 337.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

Subsequence of A002645 and hence of A028916. A006686 is a subsequence.

Take[Select[Flatten[Table[x^4+y^8,{x,40},{y,40}]],PrimeQ]//Union,40] (* Harvey P. Dale, May 01 2025 *)

list(lim)=my(v=List([2]),x4,t); for(x=1, sqrtnint(lim\=1,4), x4=x^4; forstep(y=x%2+1, sqrtnint(lim-x4,8), 2, if(isprime(t=x4+y^8), listput(v, t)))); Set(v)

A349900 Primes of the form x^2 + (y^2+1)^2.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 53, 61, 89, 101, 109, 149, 173, 181, 197, 229, 257, 269, 281, 293, 349, 353, 389, 401, 433, 461, 509, 541, 577, 601, 613, 677, 701, 733, 757, 773, 797, 809, 829, 941, 1049, 1061, 1093, 1117, 1181, 1229, 1297, 1301

Offset: 1

Charles R Greathouse IV, Jan 11 2022

nonn

Merikoski proved that there are infinitely many primes of this form, and that the order of growth of the sequence up to x is x^(3/4)/log x. (His method did not provide enough Type II information to prove that there is a C such that there are ~ C*x^(3/4)/log x.)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jori Merikoski, The polynomials X^2+(Y^2+1)^2 and X^2+(Y^3+Z^3)^2 also capture their primes, arXiv:2112.03617 [math.NT], 2021.

Subsequence of A002144.

Cf. A028916, A350687.

list(lim)=my(v=List()); lim\=1; for(y=0,sqrtint(sqrtint(lim-1)-1), my(t=(y^2+1)^2); forstep(x=2-y%2,sqrtint(lim-t),2, my(p=x^2+t); if(isprime(p), listput(v,p)))); Set(v)

is(n)=if(n<5 || !isprime(n), return(0)); for(y=0,sqrtint(sqrtint(n-1)-1), if(isprime(n-(y^2+1)^2), return(1))); 0

a(n) ≍ (n log n)^(4/3).

A350687 Primes of the form x^2 + (y^3 + z^3)^2 with x,y,z > 0.

Original entry on oeis.org

5, 13, 29, 53, 97, 173, 181, 229, 257, 277, 281, 293, 337, 617, 733, 757, 809, 881, 953, 1009, 1093, 1097, 1217, 1229, 1237, 1289, 1373, 1409, 1481, 1549, 1709, 1777, 1801, 1873, 1901, 2017, 2029, 2153, 2213, 2281, 2381, 2521, 2633

Offset: 1

Charles R Greathouse IV, Jan 11 2022

nonn

Merikoski proved that there are infinitely many primes of this form.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jori Merikoski, The polynomials X^2+(Y^2+1)^2 and X^2+(Y^3+Z^3)^2 also capture their primes, arXiv:2112.03617 [math.NT], 2021.

Subsequence of A002144.

Cf. A349900, A028916, A003325.

listA003325(lim)=my(v=List()); lim\=1; for(x=1, sqrtnint(lim-1, 3), my(x3=x^3); for(y=1, min(sqrtnint(lim-x3, 3), x), listput(v, x3+y^3))); Set(v)
list(lim)=lim\=1; my(v=List(),u=apply(sqr, listA003325(sqrtint(lim-1)))); for(x=1,sqrtint(lim-1), my(x2=x^2); for(i=1,#u, my(t=x2+u[i]); if(t>lim, break); if(isprime(t), listput(v,t)))); Set(v)

A212287 Primes of the form m*p^2 + 1, where p is prime and m <= p^2.

Original entry on oeis.org

5, 13, 17, 19, 37, 73, 101, 151, 197, 251, 401, 491, 601, 677, 727, 883, 1373, 1453, 1471, 1667, 2029, 2179, 2663, 3389, 3469, 3631, 3719, 4057, 4357, 4733, 5477, 6359, 6761, 7019, 8093, 8713, 8837, 9127, 9439, 9803, 9923, 10093, 10141, 10831, 10891, 11617, 11831, 12101, 12343

Offset: 1

Charles R Greathouse IV, Jun 13 2012

nonn

Not known to be infinite, but see the Matomäki link.

			13 is a member since 13 = 3 * 2^2 + 1 with 3 <= 2^2 and 3 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, A note on primes of the form p = aq^2 + 1, Acta Arith. 137 (2009), pp. 133-137.

Cf. A028916, A173587.

list(lim)=my(v=List(),t);lim=lim\1-.5;forprime(p=2,sqrt(lim), for(a=1,min(lim\p^2,p^2),if(isprime(t=a*p^2+1),listput(v,t))));vecsort(Vec(v),,8)

A344174 Number of primes p of the form x^4 + y^2 with y + 1 prime such that p is smaller than prime(n) and also a quadratic nonresidue modulo prime(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 3, 1, 3, 4, 4, 4, 3, 4, 2, 4, 3, 1, 3, 5, 4, 5, 3, 3, 3, 3, 3, 5, 4, 2, 4, 2, 3, 3, 3, 3, 5, 4, 5, 4, 8, 3, 4, 4, 2, 6, 5, 4, 6, 5, 8, 3, 4, 3, 5, 3, 3, 6, 4, 6, 4, 4, 4, 3, 5, 4, 7, 6, 3, 6, 5, 4, 7, 4, 5, 2, 4, 6, 2, 4, 6, 7

Offset: 1

Zhi-Wei Sun, May 10 2021

nonn

In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes of the form x^4 + y^2 with x and y integers.
Conjecture: (i) a(n) > 0 for all n > 1. In other words, for each odd prime p, there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic nonresidue modulo p.
(ii) For any odd prime p not among 3, 5, 13, 37, 277,  there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic residue modulo p.
Part (i) of the conjecture verified for all odd primes p < 2*10^9.
We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a primitive root modulo p.
See also A344173 for a similar conjecture.

			a(2) = 1, and the prime 1^4 + (2-1)^2 = 2 is a quadratic nonresidue modulo prime(2) = 3.
a(4) = 1, and the prime 1^4 + (3-1)^2 = 5 is a quadratic nonresidue modulo prime(4) = 7.
a(24) = 1, and the prime 1^4 + (7-1)^2 = 37 is a quadratic nonresidue modulo prime(24) = 89.
a(36) = 1, and the prime 1^4 + (11-1)^2 = 101 is a quadratic nonresidue modulo prime(36) = 151.
a(204) = 5, and the primes 3^4 + (11-1)^2 = 181, 3^4 + (17-1)^2 = 337, 5^4 + (5-1)^2 = 641, 5^4 + (17-1)^2 = 881 and 5^4 + (23-1)^2 = 1109 are all quadratic nonresidues modulo prime(204) = 1249.

J. Friedlander and H. Iwaniec, The polynomial x^2 + y^4 captures its primes, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.

Cf. A000040, A000290, A000583, A028916, A281792, A344173.

tab={0};Do[p:=p=Prime[n];tt={};Do[If[PrimeQ[b+1]&&PrimeQ[a^4+b^2]&&JacobiSymbol[a^4+b^2,p]==-1,tt=Append[tt,a^4+b^2]],{a,1,(p-1)^(1/4)},{b,1,(p-1-a^4)^(1/2)}];tab=Append[tab,Length[Union[tt]]],{n,2,100}];Print[tab]

A346809 Primes of the form x^2 + y^8.

Original entry on oeis.org

2, 5, 17, 37, 101, 197, 257, 281, 337, 401, 577, 617, 677, 881, 1097, 1217, 1297, 1481, 1601, 1777, 2281, 2657, 2857, 2917, 3137, 4357, 4481, 5297, 5477, 5881, 6577, 6661, 6961, 7057, 7237, 7481, 7717, 8101, 8161, 8537, 8677, 8837, 9281, 9697, 10457, 10657, 12037

Offset: 1

Michel Marcus, Aug 05 2021

nonn

Merikoski proves that this sequence is infinite.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Jori Merikoski, Exceptional characters and prime numbers in sparse sets, arXiv:2108.01355 [math.NT], 2021.

Cf. A002496 (a subsequence), A028916.

lista(lim)=my(v=List([2]), t); for(a=1, sqrtint(lim), forstep(b=a%2+1, sqrtnint(lim-a^2, 8), 2, t=a^2+b^8; if(isprime(t), listput(v, t)))); vecsort(Vec(v), , 8); \\ after A028916

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A281792 Primes of the form x^2 + p^4 where x > 0 and p is prime.

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A291206 Semi-octavan primes: primes of the form x^4 + y^8.

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A349900 Primes of the form x^2 + (y^2+1)^2.

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A350687 Primes of the form x^2 + (y^3 + z^3)^2 with x,y,z > 0.

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A212287 Primes of the form m*p^2 + 1, where p is prime and m <= p^2.

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A344174 Number of primes p of the form x^4 + y^2 with y + 1 prime such that p is smaller than prime(n) and also a quadratic nonresidue modulo prime(n).

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A346809 Primes of the form x^2 + y^8.

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