Re: Discrete hyperbolic problem

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#1237

From: Tapio Hurme
Date: Thu, 03 Jun 1999 00:00

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 (x-y)(x+y)=z , if x-y=1 then z=x+y -> y+1+y=z  and 2y=z-1 and y=(z-1)/2.
(Ref. Euler and Fermat: "Every prime is The difference of two squares".

Tapio

Paul Nulsen wrote in message ...
|Ricky Laksanadjaja <lricky@earthcorp.com> writes:
|>Hi,
|>Does anybody know if there is a fast algorithm to solve the following
|>problem?
|>Equation:  x^2 - y^2 = z  , where  x is a positive integer, y is a
|>positive integer, and  x > y, and z is a large positive integer (>200
|>digit number).  For a given Zt, can you find Xn, where Xn is the first
|>integer which is greater or equal to SQRT(Zt), i.e. Xn >= SQRT(Zt),
|>which gives an integer Ym, which will satisfy the above hyperbolic
|>equation?
(snip)
|Since x^2 - y^2 = (x + y)(x - y), any solution gives a factorization of z.
|So, if z = p q with p > q, then x = (p + q)/2 and y = (p - q)/2 give a
|solution (not necessarily integral).

(snip)

|Paul Nulsen
|

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From: "Tapio Hurme" <hurmecom@dlc.fi>
Subject: Re: Discrete hyperbolic problem
Date: 1999/06/03
Message-ID: <7j4up6$ghd$1@tron.sci.fi>
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