A proof... related to the Erdos-Strauss conjecture Rev. 9/1/99
VOCABULARY A "unit fraction" is 1/N for some positive integer N. A "positive" polynomial is a polynomial in Z[x]
whose leading coefficient is a positive integer. A "unit rational function" (URF) is a rational function
of the form 1/f(x) where f is a positive polynomial.
THEOREM Suppose that K is a positive integer, P = 4K-1, W and X
are positive integer divisors of K, and WA + X == 0 (mod P).
Then for any positive integer N, 4/(PNx+A) is the sum of
three URF's. Moreover, if A>0 then there are positive polynomials
f(x), g(x), and h(x) such that and f(0), g(0), and h(0) are positive integers. (In
addition, all of the coefficients of f, g, and h are
nonnegative.)
PROOF WLOG, we may assume that W and X are relatively prime.
(One can replace W by W/G and X by X/G, where G = GCD(W,X),
and retain the hypotheses... using the fact that G is
relatively prime to P.) Since W and X are relatively prime there is a positive
integer D such that DWX = K. Since WA + X == 0 (mod P), let
R be the integer with WA + X = PR; note that if A > 0
then R > 0. Let f(x) = K(PNx+A), g(x) = DX(WNx+R), and h(x) =
DW(PNx+A)(WNx+R). f, g, and h are positive polynomials,
and 1/g(x) + 1/h(x) 1/(DX(WNx+R)) + 1/(DW(PNx+A)(WNx+R)) (W(PNx+A) + X) / (DWX(PNx+A)(WNx+R)) (PR + WPNx) / (K(PNx+A)(WNx+R)) (P(WNx+R)) / (K(PNx+A)(WNx+R)) P / (K(PNx+A)) 4/(PNx+A) - 1/f(x) 4/(PNx+A) - 1/(K(PNx+A)) (4K-1) / (K(PNx+A)) = P / (K(PNx+A)) 1/g(x) + 1/h(x). Moreover, if A > 0 then f(0) = KA, g(0) = DXR, and
h(0) = DWAR are all positive, and all coefficients of f, g,
and h are nonnegative. QED
Taking x = 0 in the Theorem provides the result in the main page,
paraphrased as follows:
COROLLARY Suppose that A is a positive integer, and there is a
positive integer K and positive divisors W and X of K such
that Then 4/A is the sum of three unit fractions.