OFFSET
1,1
COMMENTS
Conjecture: The sequence has infinitely many terms.
Clearly all the terms must be odd and not divisible by 5. Note also that 2*(n^2+n+1)^2 - 1 = n^4 + (n+1)^4.
See also A343917 for a similar conjecture.
LINKS
Robert Israel, Table of n, a(n) for n = 1..112 (all terms < 10^10; first 53 terms from Zhi-Wei Sun)
EXAMPLE
a(1) = 71, and 2*71^2 - 1 = 10^4 + 3^4 with |10 - 3| > 1.
a(53) = 99532937, and 2*99532937^2 - 1 = 19813611095691937 = 11337^4 + 7576^4 with |11337 - 7576| > 1.
MAPLE
N:= 10^18: # for all terms <= sqrt(N)
R:= {}: count:= 0:
for x from 1 while 2*x^4 < 2*N-1 do
for y from x+3 by 2 do
v:= (x^4 + y^4 + 1)/2;
if v > N then break fi;
if issqr(v) then
m:= sqrt(v);
if not member(m, R) then
count:= count+1; R:= R union {m};
fi fi
od od:
sort(convert(R, list)); # Robert Israel, May 04 2021
MATHEMATICA
QQ[n_]:=IntegerQ[n^(1/4)];
n=0; Do[Do[If[QQ[2*m^2-1-(2x)^4]&&Abs[2x-(2*m^2-1-(2x)^4)^(1/4)]>1, n=n+1; Print[n, " ", m]; Goto[aa]], {x, 0, ((2m^2-1)^(1/4))/2}]; Label[aa], {m, 1, 25000000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 03 2021
STATUS
approved