A003828 -id:A003828 - cp's OEIS frontend

A365688 Primitive solutions k to k^2 = u^4 + v^4 + w^4, with u, v, w > 0.

481, 24961, 28721, 65441, 69121, 113241, 345761, 362401, 384161, 530881, 620321, 854401, 882889, 909321, 1094481, 1163249, 1305281, 1697761, 1855841, 2074281, 2294681, 2423601, 2568369, 2576641, 2619281, 2665721, 2696161, 2751489, 2997761, 3151281

Offset: 1

Jud McCranie, Sep 16 2023

nonn

A solution is primitive if gcd(u,v,w) = 1.
Multiplying k by a positive square gives the terms in A365657.
A term in this sequence is a square iff its square root is in A003828. The smallest term in A003828 is 422481, so the smallest square in this sequence is 422481^2 = 178490195361. - Jon E. Schoenfield, Sep 24 2023
From David A. Corneth, Sep 26 2023: (Start)
If k^2 = u^4 + v^4 + w^4 then k^2 - u^4 = (k - u^2)(k + u^2) = v^4 + w^4. Hence to find terms we can iterate over v and w to find values v^4 + w^4 which we then factor into pairs (d, t) such that d*t = (k - u^2)(k + u^2).
It follows that d and t must be even and one of (k - u^2) and (k + u^2) is NOT divisible by 4. Dividing both by 2 gives one of them odd so we only care about odd divisors of (w^4 + v^4)/4. (End)
From Jon E. Schoenfield, Sep 28 2023: (Start)
For every integer j, j^4 mod 16 = 0 if j is even, 1 if j is odd. Consequently, if k were even (which would make k^2 divisible by 4), then u,v,w would all have to be even as well, so the solution (k,u,v,w) would not be primitive. Thus every term k is odd, so k^2 mod 8 = 1, so exactly one of u,v,w is odd, and since (u^4 + v^4 + w^4) mod 16 = 1, k^2 mod 16 = 1, so k mod 8 is either 1 or 7 (not 3 or 5, because those would give k^2 mod 16 = 9).
Similarly, for every integer j, j^4 mod 5 = 0 if 5 divides j, 1 otherwise, so if k were divisible by 5 (and k^2 were thus also divisible by 5), u,v,w would all likewise have to be divisible by 5, so the solution (k,u,v,w) would not be primitive. Thus no term k is divisible by 5, so k^2 mod 5 is never 0. This leaves the only possible values of k^2 mod 5 as 1 (when k mod 5 is 1 or 4) and 4 (when k mod 5 is 2 or 3). But k^2 mod 5 must equal (u^4 + v^4 * w^4) mod 5, so k^2 mod 5 cannot be 4; it must be 1, so k mod 5 must be 1 or 4, and exactly one of u,v,w is not divisible by 5.
Thus k mod 40 = 1, 9, 31, or 39; exactly two of u,v,w are even; and exactly two of u,v,w are divisible by 5.
Conjectures:
(1) k mod 8 = 1 (hence k mod 40 is 1 or 9).
(2) Of u,v,w, the two even numbers are divisible by 4. (End)

			481^2 = 231361 = 12^4 + 15^4 + 20^4.

Jud McCranie, Table of n, a(n) for n = 1..1093

Cf. A003828, A365657.

A138760 Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.

Original entry on oeis.org

5491, 10982, 16473, 21964, 27455, 32946, 38437, 43928, 49419, 51361, 54910, 60401, 65892, 71383, 76874, 82365, 87856, 93347, 98838, 102722, 104329, 109820, 115311, 120802, 126293, 131784, 137275, 142766, 148257, 153748, 154083, 159239, 164730

Offset: 1

Jonathan Sondow, Mar 28 2008

nonn

Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).

			5491^4 = 5400^4 + (-2634)^4 + 1770^4 + 955^4 and 5491 = 5400 - 2634 + 1770 + 955, so 5491 is a member (Brudno).
51361^4 = 48150^4 + (-31764)^4 + 27385^4 + 7590^4 and 51361 = 48150 - 31764 + 27385 + 7590, so 51361 is a member (Wroblewski).
1347505009^4 = 1338058950^4 + (-89913570)^4 + 504106884^4 + (-404747255)^4, and 1347505009 = 1338058950 - 89913570 + 504106884 - 404747255, so 1347505009 is a member (Jacobi-Madden).

Simcha Brudno, A further example of A^4 + B^4 + C^4 + D^4 = E^4, Proc. Camb. Phil. Soc. 60 (1964) 1027-1028.
Noam Elkies, On A^4 + B^4 + C^4 = D^4, Math. Comp. 51 (1988) 825-835.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4, Amer. Math. Monthly 115 (2008) 220-236.
Lee W. Jacobi and Daniel J. Madden, On a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4
Eric Weisstein's MathWorld, Diophantine equation - 4th powers
Jaroslaw Wroblewski, Exhaustive list of 1009 solutions to (4,1,4) below 222,000

Cf. A003294, A003828, A096739.

n^4 = a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 with abcd =/= 0.

A175598 Numbers k such that k^4 = x^4 - y^4 - z^4.

Original entry on oeis.org

95800, 217519, 414560, 2682440, 15365639, 18796760

Offset: 1

Juri-Stepan Gerasimov, Jul 21 2010, Jul 24 2010

The values of x are 422481, 20615673, ... (without repetition).

			a(1) = 95800 because 95800^4 = 422481^4 - 217519^4 - 414560^4.
a(2) = 217519 because 217519^4 = 422481^4 - 95800^4 - 414560^4.

Cf. A003828, A175610.

First 0 removed by Jinyuan Wang, Feb 20 2020

A239247 Numbers n such that n^4 can be written as a sum of five distinct positive 4th powers.

Original entry on oeis.org

15, 30, 35, 45, 55, 60, 65, 70, 75, 85, 89, 90, 95, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 165, 170, 175, 178, 180, 185, 190, 195, 205, 210, 215, 220, 225, 230, 233, 235, 240, 245, 250, 255, 260, 265, 267, 270, 275, 280, 285, 290, 295, 300

Offset: 1

Michel Marcus, Mar 13 2014

nonn

Every multiple of a term is a term.

			15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.
35^4 = 4^4 + 21^4 + 22^4 + 26^4 + 28^4.
55^4 = 2^4 + 13^4 + 16^4 + 44^4 + 48^4.
65^4 = 1^4 + 8^4 + 12^4 + 32^4 + 64^4.
85^4 = 2^4 + 13^4 + 32^4 + 34^4 + 84^4.
89^4 = 10^4 + 35^4 + 52^4 + 60^4 + 80^4.
95^4 = 6^4 + 48^4 + 66^4 + 67^4 + 78^4.
115^4 = 4^4 + 31^4 + 48^4 + 58^4 + 112^4.
125^4 = 8^4 + 11^4 + 26^4 + 84^4 + 118^4.
145^4 = 2^4 + 23^4 + 46^4 + 52^4 + 144^4.
155^4 = 6^4 + 39^4 + 88^4 + 96^4 + 144^4.
185^4 = 2^4 + 38^4 + 62^4 + 87^4 + 182^4.
205^4 = 4^4 + 133^4 + 142^4 + 146^4 + 156^4.
215^4 = 4^4 + 26^4 + 127^4 + 174^4 + 176^4.
233^4 = 40^4 + 65^4 + 94^4 + 150^4 + 220^4.
235^4 = 9^4 + 52^4 + 148^4 + 184^4 + 194^4.

Lars Blomberg, Table of n, a(n) for n = 1..7560 (terms < 10000)
Lars Blomberg, Primitive solutions for terms < 10000. Only the first solution found for each term is included.
Eric Weisstein's World of Mathematics, Diophantine Equation 4th Powers.

Cf. A130022, A003828 (three 4th powers), A096739 (four 4th powers).

isok(n) = {ret = 0; for (x=1, sqrtnint(n^4\5, 4), for (y=x+1, sqrtnint((n^4 - x^4)\4, 4), for (z=y+1, sqrtnint((n^4 - x^4 - y^4)\3, 4), for (t=z+1, sqrtnint((n^4 - x^4 - y^4 - z^4)\2, 4), for (u=t+1, sqrtnint((n^4 - x^4 - y^4 - z^4 - t^4), 4), if (x^4+y^4+z^4+t^4+u^4 == n^4, print(n, ": ", x, ", ", y, ", ",z ,", ",t, ", ",u); ret = 1;);););););); return (ret);}

a(1) = A130022(4).

Missing terms 15 and its multiples found by Alois P. Heinz, Mar 14 2014
More examples from Michel Marcus, Mar 18 2014
More terms from Lars Blomberg, Apr 05 2014

A264764 a(n) = smallest positive k such that k^n is the sum of n-1 positive n-th powers, or 0 if no solution exists.

Original entry on oeis.org

1, 0, 422481, 144

Offset: 2

Arkadiusz Wesolowski, Nov 23 2015

a(3) = 0 (as is the case under Fermat's last theorem).

Is the number of zero terms finite?

			1^2 = 1^2.
414560^4 + 217519^4 + 95800^4 = 422481^4.
27^5 + 84^5 + 110^5 + 133^5 = 144^5.

Wikipedia, Fermat's Last Theorem
Wikipedia, Lander, Parkin, and Selfridge conjecture

Cf. A003828.

A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

Original entry on oeis.org

744, 1686, 1921, 2087, 3447, 4097, 6065, 7157, 7864, 8570

Offset: 1

Jianing Song, Jan 24 2020

Primitive solutions means gcd(a,b,c,d,e) = 1.
These are all terms from James Waldby link, which gives all solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 where k < 10000, gcd(a,b,c,d,e) = 1 and at least two of a,b,c,d,e are nonzero.
Note that if nonprimitive solutions were allowed (where at least two of a,b,c,d,e are nonzero), then 144 would be a term because 144^5 = 0^5 + 27^5 + 84^5 + 110^5 + 133^5 = 38^5 + 86^5 + 92^5 + 94^5 + 134^5.

			Solutions to k^5 = a^5 + b^5 + c^5 + d^5 + e^5 = a'^5 + b'^5 + c'^5 + d'^5 + e'^5:
744: (100, 210, 414, 629, 651), (14, 95, 545, 586, 644);
1686: (265, 486, 784, 791, 1670), (46, 591, 675, 999, 1655);
1921: (275, 351, 872, 1298, 1855), (95, 771, 1020, 1519, 1756);
2087: (145, 565, 1105, 1462, 1990), (519, 642, 1026, 1480, 1990);
3447: (1212, 1300, 1345, 1699, 3411), (289, 317, 1033, 1682, 3426);
4097: (1281, 2154, 2396, 3462, 3504), (954, 1989, 2127, 2396, 3981);
6065: (3629, 3811, 4070, 4272, 5313), (854, 3160, 3752, 5073, 5196);
7157: (1827, 2186, 4789, 5629, 6376), (930, 2746, 3570, 5109, 6802);
7864: (1093, 2309, 3629, 6137, 7296), (312, 1631, 3418, 3544, 7809);
8570: (1766, 2529, 4086, 5520, 8319), (2101, 2315, 2710, 3960, 8524).

James Waldby, A Table of Fifth Powers equal to a Fifth Power

Subsequence of A063923 (and thus of A063922).
Other similar sequences:
A023041 (k^3=a^3+b^3+c^3, gcd(a,b,c)=1);
A003828 (k^4=a^4+b^4+c^4, gcd(a,b,c)=1);
A175610 (k^4=a^4+b^4+c^4);
A039664 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1);
A003294 (k^4=a^4+b^4+c^4+d^4);
A331675 (k^4=a^4+b^4+c^4+d^4, gcd(a,b,c,d)=1, at least two solutions).
A134341 (k^5=a^5+b^5+c^5+d^5).

A365657 Integers k such that k^2 can be written as the sum of three positive fourth powers.

Original entry on oeis.org

481, 1924, 4329, 7696, 12025, 17316, 23569, 24961, 28721, 30784, 38961, 48100, 58201, 65441, 69121, 69264, 81289, 94276, 99844, 108225, 113241, 114884, 123136, 139009, 155844, 173641, 192400, 212121, 224649, 232804, 254449, 258489, 261764, 276484, 277056, 300625, 325156

Offset: 1

Jud McCranie, Sep 14 2023

nonn

Primitive solutions are in A365688.
From Jon E. Schoenfield, Sep 15 2023: (Start)
If k is a term, then so is m^2 * k for every m > 1.
Every even term is four times a smaller term.
Every odd term is the square root of the sum of one odd fourth power and two even fourth powers.
(End)

			1924^2 = 24^4 + 30^4 + 40^4.

Jean-Marie De Koninck, "Those Fascinating Numbers", AMS, 2008, entry 481.

Cf. A003828, A365688.

from itertools import count, islice
from sympy import integer_nthroot
def A365657_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        m, flag = k**2, False
        for x in count(1):
            if (x4:=x**4)+2>m or flag:
                break
            for y in range(min(x,integer_nthroot(m-x4-1,4)[0]),0,-1):
                if (z4:=m-x4-(y4:=y**4))>y4 or flag:
                    break
                if integer_nthroot(z4,4)[1]:
                    yield k
                    flag = True
                    break
A365657_list = list(islice(A365657_gen(),6)) # Chai Wah Wu, Sep 19 2023

A121043 Denominators of points on x^4+y^4+z^4=353.

Original entry on oeis.org

1, 3, 583, 787, 3269, 17057, 17461

Offset: 1

Thomas Womack (tom(AT)womack.net), Sep 06 2006

Next term is at least 151814 since I've searched x^4+y^4+z^4 for x,y,z<500009

			43^4 + 956^4 + 2514^4 = 353*583^4

Cf. A003828.

cp's OEIS Frontend

A365688 Primitive solutions k to k^2 = u^4 + v^4 + w^4, with u, v, w > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A138760 Numbers n such that n^4 is a sum of 4th powers of four nonzero integers whose sum is n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A175598 Numbers k such that k^4 = x^4 - y^4 - z^4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A239247 Numbers n such that n^4 can be written as a sum of five distinct positive 4th powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A264764 a(n) = smallest positive k such that k^n is the sum of n-1 positive n-th powers, or 0 if no solution exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A331674 Numbers k such that k^5 = a^5 + b^5 + c^5 + d^5 + e^5 has at least two primitive solutions in nonnegative integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A365657 Integers k such that k^2 can be written as the sum of three positive fourth powers.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Python

A121043 Denominators of points on x^4+y^4+z^4=353.

Views

Author

Keywords

Comments

Examples

Crossrefs