A028916 -id:A028916 - cp's OEIS frontend

A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

5, 13, 29, 41, 53, 61, 73, 89, 109, 113, 137, 149, 157, 173, 193, 229, 233, 269, 281, 293, 313, 317, 349, 353, 373, 389, 397, 409, 433, 449, 461, 509, 521, 557, 569, 593, 601, 613, 617, 653, 673, 701, 733, 761, 773, 797, 809, 853, 857, 877, 929, 937, 941, 953

Offset: 1

Charles R Greathouse IV, Feb 18 2011

Sequence is infinite, see Fouvry & Iwaniec.
Its intersection with A028916 is A262340, by the uniqueness part of Fermat's two-squares theorem. - Jonathan Sondow, Oct 05 2015
Named after the French mathematician Étienne Fouvry (b. 1953) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 20 2021

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica, Vol. 79, No. 3 (1997), pp. 249-287.
Lasse Grimmelt, Vinogradov's Theorem with Fouvry-Iwaniec Primes, arXiv:1809.10008 [math.NT], 2018.
Art of Problem Solving, Fermat's Two Squares Theorem.

Subsequence of A002144 and hence of A002313.
The positive terms of A240130 form a subsequence.
Cf. A010052, A001248, A000040, A028916, A262340.

a185086 n = a185086_list !! (n-1)
a185086_list = filter (\p -> any ((== 1) . a010052) $
               map (p -) $ takeWhile (<= p) a001248_list) a000040_list
-- Reinhard Zumkeller, Mar 17 2013

nn = 1000; Union[Reap[Do[n = k^2 + p^2; If[n <= nn && PrimeQ[n], Sow[n]], {k, Sqrt[nn]}, {p, Prime[Range[PrimePi[Sqrt[nn]]]]}]][[2, 1]]]

is(n)=forprime(p=2,sqrtint(n),if(issquare(n-p^2),return(isprime(n))));0

list(lim)=my(v=List(),N,t);forprime(p=2,sqrt(lim), N=p^2; for(n=1,sqrt(lim-N), if(ispseudoprime(t=N+n^2), listput(v,t)))); v=vecsort(Vec(v),,8); v

A256852 Number of ways to write prime(n) = a^2 + b^4.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

a(A049084(A028916(n))) > 0; a(A049084(A256863(n))) = 0;
Conjecture: a(n) <= 2, empirically checked for the first 10^6 primes.
The conjecture is true, because by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, if a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. - Jonathan Sondow, Oct 03 2015
Friedlander and Iwaniec proved that a(n) > 0 infinitely often. - Jonathan Sondow, Oct 05 2015

			First numbers n, such that a(n) > 0:
.   k |  n |   prime(n)                    | a(n)
. ----+----+-------------------------------+-----
.   1 |  1 |    2 = 1^2 + 1^4              |   1
.   2 |  3 |    5 = 2^2 + 1^4              |   1
.   3 |  7 |   17 = 1^2 + 2^4 = 4^2 + 1^4  |   2
.   4 | 12 |   37 = 6^2 + 1^4              |   1
.   5 | 13 |   41 = 5^2 + 2^4              |   1
.   6 | 25 |   97 = 4^2 + 3^4 = 9^2 + 2^4  |   2
.   7 | 33 |  101 = 10^2 + 1^4             |   1
.   8 | 42 |  181 = 10^2 + 3^4             |   1
.   9 | 45 |  197 = 14^2 + 1^4             |   1
.  10 | 53 |  241 = 15^2 + 2^4             |   1
.  11 | 55 |  257 = 1^2 + 4^4 = 16^2 + 1^4 |   2
.  12 | 59 |  277 = 14^2 + 3^4             |   1
.  13 | 60 |  281 = 5^2 + 4^4              |   1
.  14 | 68 |  337 = 9^2 + 4^4 = 16^2 + 3^4 |   2
.  15 | 79 |  401 = 20^2 + 1^4             |   1
.  16 | 88 |  457 = 21^2 + 2^4             |   1 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS, vol. 94 no. 4, pp. 1054-1058.
Wikipedia, Friedlander-Iwaniec theorem

Cf. A000040, A000290, A000583, A010052, A002645, A028916, A256863.

a256852 n = a256852_list !! (n-1)
a256852_list = f a000040_list [] $ tail a000583_list where
   f ps'@(p:ps) us vs'@(v:vs)
     | p > v     = f ps' (v:us) vs
     | otherwise = (sum $ map (a010052 . (p -)) us) : f ps us vs'

A262340 Primes of the form p^2 + b^4 where p is a prime.

Original entry on oeis.org

5, 41, 137, 281, 617, 857, 977, 1097, 1217, 1321, 1657, 1697, 2137, 4217, 4457, 4937, 5297, 6257, 6337, 7537, 7577, 7817, 7937, 9137, 10009, 10169, 10289, 10337, 10457, 10529, 11369, 11497, 11681, 11897, 12809, 13177, 13721, 14489, 15329, 16889, 17417

Offset: 1

Jonathan Sondow, Oct 03 2015

nonn

It is not known whether there are infinitely many primes of such form.

Same as the intersection of A185086 (primes of the form p^2 + k^2 where p is a prime) with A028916 (primes of the form a^2 + b^4). (Proof: Clearly, p^2 + b^4 is in A185086 and in A028916. Conversely, if a(n) = p^2 + k^2 = a^2 + b^4, then by the uniqueness part of Fermat's two squares (or 4n+1) theorem, (p,k) = (a,b^2) or (p,k) = (b^2,a). But the latter is impossible since p is prime, so a(n) = p^2 + b^4.)

			5 = 2^2 + 1^4, so a(1) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem
MathWorld, Fermat's 4n+1 Theorem

Cf. A028916, A185086.

nn = 14; Union[ Flatten[ Table[ Select[ Prime[n]^2 + Range[nn]^4, PrimeQ[#] && # < nn^4 &], {n,PrimePi[nn^2]}]]]

list(lim)=my(v=List(),p2,t); forprime(p=2,sqrtint(lim\=1), p2=p^2; forstep(x=1+p%2,sqrtnint(lim-p2,4),2, if(isprime(t=p2+x^4), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 21 2017

A078523 Primes of the form a^2 + b^6.

Original entry on oeis.org

2, 5, 17, 37, 73, 89, 101, 113, 197, 233, 257, 353, 401, 577, 593, 677, 733, 829, 1129, 1153, 1213, 1289, 1297, 1433, 1601, 1753, 1913, 2089, 2273, 2917, 3089, 3137, 3229, 3313, 3433, 4093, 4177, 4217, 4289, 4357, 4457, 4721, 4937, 5393, 5477, 5689, 6121

Offset: 1

T. D. Noe, Nov 26 2002

Friedlander and Iwaniec prove that there are an infinite number of primes of the form a^2+b^4 (A028916). They speculate that the a^2+b^6 case can be proved by similar methods.

			73 = 3^2 + 2^6

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS February 18, 1997 94 (4) 1054-1058.
Jori Merikoski, A Cubic analogue of the Friedlander-Iwaniec spin over primes, arXiv:2012.05675 [math.NT], 2020.

Cf. A028916.

maxN=10000; lst={}; Do[p=i^2+j^6; If[p

list(lim)=my(v=List([2]),b6,t); lim\=1; for(b=1,sqrtnint(lim-1,6), b6=b^6; forstep(a=1+b%2,sqrtint(lim-b6),2, if(isprime(t=a^2+b6), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 18 2017

A100271 Primes of the form a^4 + b^3 with b>0.

Original entry on oeis.org

2, 17, 43, 89, 257, 283, 359, 593, 599, 1297, 2213, 2617, 3391, 3631, 4129, 4177, 4721, 6569, 7561, 8081, 8233, 9277, 10343, 10657, 10729, 11273, 12197, 13049, 13463, 14449, 14561, 15641, 15881, 16369, 16921, 17209, 17657, 19699, 22067, 24137

Offset: 1

T. D. Noe, Nov 18 2004

nonn

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

Cf. A002645 (primes of the form a^4 + b^4), A028916 (primes of the form a^4 + b^2), A100291 (numbers of the form a^4 + b^3).

lst={}; Do[p=a^4+b^3; If[p<50000&&PrimeQ[p], AppendTo[lst, p]], {a, 64}, {b, 256}]; Union[lst]

list(lim)=my(v=List([2]),a4,t); lim\=1; for(a=1,sqrtnint(lim-1,4), a4=a^4; forstep(b=1+a%2,sqrtnint(lim-a4,3),2, if(isprime(t=a4+b^3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 18 2017

A247857 Primes of the form a^2 + b^4, with repetition.

Original entry on oeis.org

2, 5, 17, 17, 37, 41, 97, 97, 101, 137, 181, 197, 241, 257, 257, 277, 281, 337, 337, 401, 457, 577, 617, 641, 641, 661, 677, 757, 769, 821, 857, 881, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2417, 2437

Offset: 1

Jean-François Alcover, Sep 25 2014

nonn

Duplicates, which begin 17, 97, 257, 337, etc, are quartan primes A002645, except 2 (noticed by Michel Marcus).
Is there any triple?
No, by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, when a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. (This also proves Marcus's comment, since a^2 + b^4 = b^4 + B^4.) - Jonathan Sondow, Oct 03 2015

			Since 97 = 4^2 + 3^4 = 9^2 + 2^4, it appears twice in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS, vol. 94 no. 4, pp. 1054-1058.
Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], 2010, p. 8.
Wikipedia, Friedlander-Iwaniec theorem

Cf. A002645, A028916 (same sequence without repetition).

Cf. A000040, A256852.

a247857 n = a247857_list !! (n-1)
a247857_list = concat $ zipWith replicate a256852_list a000040_list
-- Reinhard Zumkeller, Apr 11 2015

max = 10^4; r = Reap[Do[n = a^2 + b^4; If[n <= max && PrimeQ[n], Sow[n]], {a, Sqrt[max]}, {b, max^(1/4)}]][[2, 1]]; Union[r, SameTest -> (False&)]

A247858 Decimal expansion of the value of the continued fraction [0; 2, 5, 17, 17, 37, 41, 97, 97, ...], generated with primes of the form a^2 + b^4.

Original entry on oeis.org

4, 5, 5, 0, 2, 4, 8, 1, 6, 4, 9, 0, 1, 7, 0, 0, 2, 2, 3, 6, 9, 0, 5, 2, 8, 0, 8, 2, 7, 9, 7, 4, 4, 8, 2, 4, 1, 0, 5, 7, 5, 5, 5, 4, 8, 9, 0, 5, 0, 7, 6, 4, 4, 0, 5, 6, 8, 5, 4, 1, 8, 5, 9, 1, 5, 0, 8, 4, 6, 0, 8, 5, 0, 1, 0, 7, 1, 8, 6, 3, 1, 4, 3, 6, 3, 1, 0, 6, 6, 7, 6, 9, 7, 5, 4, 6, 0, 4, 5, 1, 9, 9, 2

Offset: 0

Jean-François Alcover, Sep 25 2014

			1/(2 + 1/(5 + 1/(17 + 1/(17 + 1/(37 + 1/(41 + 1/(97 + 1/(97 + ...))))))))
0.45502481649017002236905280827974482410575554890507644...

John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS February 18, 1997 94 (4) 1054-1058.
Wikipedia, Friedlander-Iwaniec theorem
Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], 2010, pp. 8-9.

Cf. A028916, A243340, A247857.

max = 1000; r = Reap[Do[n = a^2 + b^4; If[n <= max && PrimeQ[n], Sow[n]], {a, Sqrt[max]}, {b, max^(1/4)}]][[2, 1]]; u = Union[r, SameTest -> (False&)] ; RealDigits[FromContinuedFraction[Join[{0}, u]], 10, 103] // First

A256863 Primes that can't be written in form a^2 + b^4.

Original entry on oeis.org

3, 7, 11, 13, 19, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 103, 107, 109, 113, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 191, 193, 199, 211, 223, 227, 229, 233, 239, 251, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 347, 349, 353

Offset: 1

Reinhard Zumkeller, Apr 11 2015

nonn

Complement of A028916;

A256852(A049084(a(n))) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A028916, A000290, A000583, A000040, A256852.

a256863 n = a256863_list !! (n-1)
a256863_list = map a000040 $ filter ((== 0) . a256852) [1..]

Reap[Do[If[Reduce[p == a^2 + b^4, {a, b}, Integers] === False, Sow[p]], {p, Prime[Range[80]]}]][[2, 1]] (* Jean-François Alcover, Jan 31 2018 *)

A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

Original entry on oeis.org

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.

The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]

do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

a(n)^2 = A185086(n) - k^2 for some integer k > 0.

A276533 Least prime p with A271518(p) = n.

Original entry on oeis.org

5, 2, 19, 127, 17, 67, 163, 41, 89, 101, 131, 313, 257, 211, 227, 461, 241, 401, 613, 337, 433, 353, 577, 467, 863, 887, 617, 787, 601, 569, 761, 641, 823, 673, 857, 1217, 881, 1091, 1289, 977, 1427, 1097, 1801, 929, 1153, 953, 1321, 1049, 1747, 1409

Offset: 1

Zhi-Wei Sun, Dec 12 2016

nonn

Conjecture: a(n) exists for any positive integer n.
In contrast, it is known that for each prime p the number of ordered integral solutions to the equation x^2 + y^2 + z^2 + w^2 = p is 8*(p+1).
In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes p of the form w^2 + x^4 = w^2 + (x^2)^2 + 0^2 + 0^2 with w and x nonnegative integers. Since x^2 + 3*0 + 5*0 is a square, we see that A271518(p) > 0 for infinitely many primes p.

			a(1) = 5 since 5 is the first prime which can be written in a unique way as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integer and x + 3*y + 5*z a square; in fact, 5 = 1^2 + 0^2 + 0^2 + 2^2 with 1 + 3*0 + 5*0 = 1^2.
a(2) = 2 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 = 1^2, and 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..500
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.
Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Cf. A000040, A000118, A000290, A028916, A271518, A273294, A273302, A278560.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[m=0;Label[aa];m=m+1;r=0;Do[If[SQ[Prime[m]-x^2-y^2-z^2]&&SQ[x+3y+5z],r=r+1;If[r>n,Goto[aa]]],{x,0,Sqrt[Prime[m]]},{y,0,Sqrt[Prime[m]-x^2]},{z,0,Sqrt[Prime[m]-x^2-y^2]}];If[r

cp's OEIS Frontend

A185086 Fouvry-Iwaniec primes: Primes of the form k^2 + p^2 where p is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

A256852 Number of ways to write prime(n) = a^2 + b^4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A262340 Primes of the form p^2 + b^4 where p is a prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A078523 Primes of the form a^2 + b^6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A100271 Primes of the form a^4 + b^3 with b>0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A247857 Primes of the form a^2 + b^4, with repetition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A247858 Decimal expansion of the value of the continued fraction [0; 2, 5, 17, 17, 37, 41, 97, 97, ...], generated with primes of the form a^2 + b^4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A256863 Primes that can't be written in form a^2 + b^4.

Views

Author

Keywords

Comments

Links