Re: Is it true ?

From: Asger Grunnet (asger_at_adslhome.dk)
Date: 10/03/04 

Date: Sun, 3 Oct 2004 12:26:59 +0200

"RICH" wrote:
> Set a,b,p in N. p is odd , one of a,b is even.a is not equal b
>
> a is bigger then 11 , b is bigger than 10
>
> If p^3=a^2 + b^2 - ab is ture , we can say :(a,b)=x ( x is bigger than
> 1 ) ?
>
> For example : 7^3 = 21^2 + 14^2 -14*21
>
> ( 21,14 )=7
>
> If ( a,b)=1 , p^3 is not equal a^2 + b^2 - ab . Is it true ?

If I understand your question correctly, you are asking whether
there exist integers a, b, p such that

(1) a > 11,
(2) b > 10,
(3) a =/= b,
(4) p^3 = a^2 + b^2 - ab.
(5) exactly one of a and b is even
(6) gcd(a, b) = 1,

(Note that (4) + (5) implies that p is odd.)

Such integers exist, for example (a, b) = (17, 90), (18, 19), (37, 360),
(38, 95), (71, 252), (73, 90), (90, 199), (126, 323), (181, 252),
(216, 703), (270, 971), (323, 360), (378, 629), ...

Notice that if (a,b) is a solution with b > a and b even, then (b-a, b) is
also a solution.

Asger.

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