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Help with programming problem: a^4+8b^4 = c^4+8d^4? (Relevant for
quasi-Waring problem)
Posted:
Nov 10, 2009 1:16 AM
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Hello all,
By Ryley's Theorem, it has long been known that any N is the sum of
three rational cubes (positive or negative). The proof is a rather
simple algebraic identity. Also, by means of an identity, A.J.
Choudhry proved than any N is the sum of 6 fifth powers of rationals,
and 8 seventh powers of rationals (On Sums of Seventh Powers; 1999).
Though he didn't spell it out, the latter result depended on solving
the simultaneous eqns:
a^2+32b^2 = c^2+32d^2 (eq.1a)
a^4+8b^4 = c^4+8d^4 (eq.1b)
He gave a 33-digit soln {a,b,c,d}. But this was a decade ago, with
slower computers, so he may have used assumptions that skipped the
smaller ones. Question 1: Anyone can find smaller solns to eq.1a and
b?
To make things easier, the general case is this:
a^2+m^5b^2 = c^2+m^5d^2 (eq.2a)
a^4+m^3b^4 = c^4+m^3d^4 (eq.2b)
for m > 1. (Note how the paired exponents add up to 7.) Choudhry
simply chose m = 2.
Question 2: If anyone can find small solns to eq.2a,b for any m > 1,
then you'll have a new proof that any number N is the sum of 8 seventh
powers of rationals.
- Titus
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