I'm going to shift variables to give you all something more familiar
with factoring and explain the significance of a remarkable and simple
congruence result:

Given

x^2 = y^2 + cN

where N is a target odd composite to factor, not divisible by 3, and c
is a control variable that equals 1 if N mod 3 = 2 and 5 if N mod 3 =
1, it can be trivially proven that

x^2 = 8^{-1] (9cN) mod p

when 8^{-1] (9cN) is a quadratic residue for an odd prime p, where p <
2x/3.

For some of you it may be an extremely counter-intuitive result or
terribly hard to understand, so I'll give an example and then talk
about why it's significant.

Let N = 299 = 13(23), and since 299 mod 3 = 2, c=1. The only primes
available (I'll leave the proof to the reader) are 3, 5, 7, 11 and 13
but since I know 13 is a factor I'll try, 11.

With p = 11, I check to see if the quadratic residue exists:

On Jun 4, 8:39 pm, JSH gmail.com> wrote:

and that Google fact is the irrefutable proof that all your proofs are
correct. Hurray for Harris. He's one of the top 10 mathematicians
ever. That's what Google says and you can't argue with Google.

Psychoanalysis teaches that we are all narcissistic at an early stage of
our lives. As infants and toddlers we all feel that we are the centre of
the Universe, the most important, omnipotent and omniscient beings. At
that phase of our development, we perceive our parents as mythical
figures, immortal and awesomely powerful but there solely to cater to
our needs, to protect and nourish us. Both Self and others are viewed
immaturely, as idealisations. This, in the psychodynamic models, is
called the phase of "primary" narcissism.

Inevitably, the inexorable conflicts of life lead to disillusionment. If
this process is abrupt, inconsistent, unpredictable, capricious,
arbitrary and intense, then the injuries sustained by the infant's...

On Jun 4, 9:20 pm, Dan Eperstein hyghway.net> wrote:

Here is another one:

http://www.engadget.com/tag/archerquinn

On 5 Jun, 01:39, JSH gmail.com> wrote:

Rotwang wrote:

On 5 Jun, 21:11, Rick Decker hamilton.edu> wrote:

Rotwang wrote:

On Jun 4, 8:39 pm, JSH gmail.com> wrote:

Do you have any other guesses?

JSH wrote:

Here it is: A mathematical proof begins with a truth and proceeds by
logical steps to a conclusion which then must be true.

Is it proven that the sum of the internal angles of a triangle is 180
degrees, or is it proven that it is less that 180 degrees?

In euclidean geometry one proves the first (= 180) and in hyperbolic
geometry the second (< 180). What has truth got to do with it in either
case?