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
KEYWORD
nonn
AUTHOR
Mohammad Tejabwala, Jun 21 2021
STATUS
approved