Prev: proof of Goldbach has to use 10^500 #801 Correcting Math
Next: sin(b)*sin(a+(1/2)*b) = sin(a)*sin((1/2)*a+b)=sin(b+a)*cos((a-b)*(1/2))
From: TPiezas on 13 Aug 2010 16:01
Hello all,

I. The "Euler brick" is well-known. These are triples (a,b,c) such
that,

a^2+b^2 = x^2
a^2+c^2 = y^2
b^2+c^2 = z^2

all in positive integers.

II. Euler also considered quadruples (a,b,c,d) such that,

a^2+b^2+c^2 = x^2
a^2+b^2+d^2 = y^2
a^2+c^2+d^2 = z^2
b^2+c^2+d^2 = t^2

Interestingly, these can be solved using Pythagorean triples, or even
the diophantine eqn p^4+q^4 = r^4+s^4.

For small positive integer solns, Jarek Wroblewski did a search and
found there are only 5 with {a,b,c,d} < 1000 and gcd(a,b,c,d) = 1,
namely,

1. {49, 72, 72, 84}
2. {21, 28, 120, 120}
3. {60, 105, 168, 280}
4. {313, 336, 336, 492}
5. {237, 336, 336, 952}

Euler found [2] and [3] using a polynomial parametrization, but missed
[1]. Can you find the polynomial family, if any, that has [1] as a
member and which Euler missed?

For more details, see http://sites.google.com/site/tpiezas/updates08

- Titus
 | 
Pages: 1
Prev: proof of Goldbach has to use 10^500 #801 Correcting Math
Next: sin(b)*sin(a+(1/2)*b) = sin(a)*sin((1/2)*a+b)=sin(b+a)*cos((a-b)*(1/2))