I think the square-free terms, namely: 17, 82, 97, 113, 193, 257, 274, 337, 433, 514, 577, 593, 626, 641, 673, 706, 881, 914, ... make a more interesting sequence, and it is already in OEIS as A184982. On Sun, Mar 4, 2012 at 6:16 PM, Neil Sloane <njasloane@gmail.com> wrote:
I added the sequence from "Search II" as A209078. Comments welcomed. Neil
On Sat, Dec 10, 2011 at 11:56 AM, Warren Smith <warren.wds@gmail.com> wrote:
Fermat in about 1640 famously proved a^4+-b^4=c^2 has no positive integer solutions. Also a^4-b^4=2*c^2 has no positive integer solutions. These results are shown in T.Nagell: Intro to number theory, Wiley 1951, pp.227-230. However, a^4+b^4=2*c^2 trivially has a ton of solutions a=b, c=a^2. If we exclude the trivial by demanding b>a>0 though, I don't know of any solutions.
Let us generalize. Ask about a^4 + b^4 = d*c^2 with a,b,c,d integers and b>a>0.
My computer looked for solutions. Search I: a^4+b^4=d*c^2 with 0<a<b<65536 and 0<d<65. Result: a ton of solutions for d=17, but zero solutions for any other d. a=1, b=2, d=17, c=1 a=2, b=4, d=17, c=4 a=2, b=13, d=17, c=41 a=3, b=6, d=17, c=9 a=4, b=8, d=17, c=16 ...
Well, that's interesting: 17 is special.
Search II: a^4+b^4=d*c^2 with 0<a<b<=1024 and 0<d<=1024. Result: The following d are "special": 17, 68, 82, 97, 113, 153, 193, 257, 272, 274, 328, 337, 388, 425, 433, 452, 514, 577, 593, 612, 626, 641, 673, 706, 738, 772, 833, 873, 881, 914, 1017. [This sequence is not in OEIS.]
What is going on here? I do not really know, but an initial speculation is this. When d=17, there is the easy solution a=1, b=2, d=17, c=1.
But here is the thing: if there is one solution, then it can usually be used to generate an infinite set of solutions. I got that from Tito Piezas's online book http://sites.google.com/site/tpiezas/018 where he attributes it to A.Desboves (probably about 1880?). So there are "special" d with an infinity of solutions, and there are "usual" d with zero solutions (and conceivably there are a few "crazy" d with a finite nonzero number of solutions because the below Desboves method for generating more solutions happens merely to regenerate old ones, which would be an amazing miracle).
How to generate more solutions from one: Piezas says Desboves said (as edited by me): if p^4+q^4 = d*r^2, then x^4+y^4 = d*z^2 where x = p*(4*m^2-3*(m+n)^2), y = q*(4*n^2-3*(m+n)^2), z = r*(4*(m+n)^4-3*(m-n)^4) where m=p^4 and n=q^4.
So for example starting from p=1, q=2, d=17, r=1 we get x=863, y=314, z=182209 and sure enough 863^4 + 314^4 = 17*182209^2 is another solution.
Questions: Q1. Can you find a parameterized family of d=17 solutions? Q2. Can you understand the set of special d? Q3. Can you prove there are no "crazy" d (or find them all)?
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