0,3

Squaring the first equation and setting the result equal to the second, we need d+b+2*sqrt(d*b)=d^2+b -> d+2*sqrt(d*b)=d^2 -> d^2-d = 2*sqrt(d*b)

-> d^2*(d-1)^2=4*d*b -> b=d*(d-1)^2/4 -> sqrt(b)=(d-1)*sqrt(d)/2. Setting d=(n+1)^2 yields sqrt(b)=A027480(n).

This is the case k=2 for FLTR, Fermat's Last Theorem with rational exponents 1/k: Consider x + y = x + y. Then (x^k)^(1/k) + (y^k)^(1/k) = ((x+y)^k)^(1/k).

For k > 2, there is an infinite number of solutions to d^(1/k) + b^(1/k) = c^(1/k). E.g. 8^(1/3) + 27^(1/3) = 125^(1/3) at k=3. However, in conjunction with d^2+b = c, I could not find any nontrivial solutions.

A shifted version of A027480. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 07 2009]

C. D. Bennet, A. M. W. Glass and G. J. Szekely, Fermat's Last Theorem for Rational Exponents, Am. Math. Monthly 111 (2004), 322-329. - R. J. Mathar, Apr 21 2009

O.g.f.: 3*x^2/(-1+x)^4 . a(n) = n(n^2-1)/2 = A007531(n+1)/2 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 20 2008

For d=9, b=144, c=225, 9^(1/2) + 144^(1/2) = 225^(1/2) and 9^2 + 144 = 225. So b^(1/2) = 12 is the 4th entry in the sequence.

with(finance):seq(add(cashflows([n*k, k, k], 0 ), k=0..n), n=-1..51); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008

(PARI) flt2(n, p) = { local(a, b); for(a=0, n, b = (a^3-a)/2; print1(b", ") ) }

Sequence in context: A164013 A057671 A027480 this_sequence A048088 A064181 A089143

Adjacent sequences: A135500 A135501 A135502 this_sequence A135504 A135505 A135506

nonn

Cino Hilliard (hillcino368(AT)hotmail.com), Feb 09 2008

Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 21 2009