A345645 Numbers whose square can be represented in exactly one way as the sum of a square and a biquadrate (fourth power).

5, 15, 20, 34, 39, 41, 45, 60, 80, 85, 111, 125, 135, 136, 150, 156, 164, 175, 180, 194, 219, 240, 245, 255, 265, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 500, 505, 514, 540, 544, 600, 605, 609, 624, 629, 656, 671, 674, 689

Offset

1,1

Comments

Numbers z such that there is exactly one solution to z^2 = x^2 + y^4.

From Karl-Heinz Hofmann, Oct 21 2021: (Start)

No term can be a square (see the comment from Altug Alkan in A111925).

Terms must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).

Additionally, if the terms have prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.

The special prime factor 2 has the same behavior, i.e., if the term is even, x and y must be even too. (End)

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Example

3^2 + 2^4 = 9 + 16 = 25 = 5^2, so 5 is a term.
60^2 + 5^4 = 63^2 + 4^4 = 65^2, so 65 is not a term.

Mathematica

Select[Range@100, Length@Solve[x^2+y^4==#^2&&x>0&&y>0, {x, y}, Integers]==1&] (* Giorgos Kalogeropoulos, Jun 25 2021 *)

Prog

(Python)
terms = []
for i in range(1, 700):
    occur = 0
    ii = i*i
    for j in range(1, i):
        k = int((ii - j*j) ** 0.25)
        if k*k*k*k + j*j == ii:
            occur += 1
    if occur == 1:
        terms.append(i)
print(terms)
(PARI) inlist(list, v) = for (i=1, #list, if (list[i]==v, return(1)));
isok(m) = {my(list = List()); for (k=1, sqrtnint(m^2, 4), if (issquare(j=m^2-k^4) && !inlist(vecsort([k^4, j^2])), listput(list, vecsort([k^4, j^2]))); ); #list == 1; } \\ Michel Marcus, Jun 26 2021

Crossrefs

Cf. A000290, A000583, A180241, A271576 (all solutions).

Cf. A345700 (2 solutions), A345968 (3 solutions), A346110 (4 solutions), A348655 (5 solutions), A349324 (6 solutions), A346115 (the least solutions).

Cf. A002144 (p == 1 (mod 4)), A002145 (p == 3 (mod 4)).

Sequence in context: A045176 A273908 A271576 * A274535 A027184 A328249

Adjacent sequences: A345642 A345643 A345644 * A345646 A345647 A345648

Keyword

nonn

Author

Mohammad Tejabwala, Jun 21 2021

Status

approved