Pell’s Equation

..

The writer admits to not knowing sufficient about Pell’s equation but
at the same time being very interested in the prospect of a solution
to the so-called factoring problem. It behoves me to put my cards on
the table quickly and admit to this fact rather than go to Wikipedia
and come back with a thin cosmetic understanding before then going on
line as a groupie pretending to be a guru. I do know however that the
number hast to be a 'natural' number and it must also be a perfect
square number.

Only a fool would try to discuss this very important project without
first of all getting his head around a full understanding of the
subject and this writer just hasn’t the time to do this at present i.e
to go deeply into into Legrange mathematics. Pell’s equation must
surely be well known in pure mathematics and is bound to be backed up
with proven theorems within mathematics so the writer is not going to
challenge any of the claims made by the recent poster. It does seem
to be too good to be true however and the writer is excited at the
prospect of it being demonstrated.

It is no doubt a contradiction to ask for a demonstration of the
recent postings if the algebraic presentation is mathematically
provable which this writer is taking unquestionably as irrefutable
fact. Such a move is a case of putting the horse before the cart –
the proof of the general formulae is suppose to obviate this – but we
are not dealing with something that depends on mathematical
continuity, instead we are testing the un-mathematical constraints of
two things – the capacity of an ordinary 32 bit computer to find the
prime factors of a number that is currently, greater than 2**31 and
the complexity of long hand working in terms of time taken to achieve
the finding of all the prime factors of a very large number.

By way of explanation, the common home computer has 32- bit
architecture => the largest positive integer number it can handle is
2**31 = 2147483648 (more recent operating systems come with 64-bit
arithmetic => 2 **63 storage is possible). If the public-key of an
RSA cipher is less than this then it is possible to decrypt the
ciphertext by means of a computer by finding the prime factor that was
used as the key basis and that is why integers of some 300+ digits
are used to prevent this happening. The upshot of course is that
there is no computer yet in existence that can store such large
integers during the factoring process. The alternative is longhand
factoring by human beings but this would take many millions of years
so clearly, the constraints are this - integer cell storage limitation
in computers and time taken in longhand methods by human beings. If a
solution to either of these problems is ever achieved, that is, an
advance in computer science that will achieve large integer storage
and high speed factoring in a computer or an improved algorithm that
will enable high speed factoring by human longhand methods, then its
curtains for the RSA cipher.

It appears to the writer that the recent posting i.e. factoring of any
integer using an adaptation of Pell’s equation is claiming to be able
to do this.

It would be absolutely great if that is true but it has to be a case
now of “the proof of the pudding is in the eating” and it is necessary
to ask the poster to factor some large number – greater that
2147483648 – as a demonstration of his work using Pell’s equation.
The writer will be happy with, say, the two largest prime factors of
the chosen number whatever it may be but all workings must be
transparent and the number should be as large as possible within
reason.

Good luck with the general project – your country needs you -
adacrypt.

PS, Proving that the number is a perfect square is a condition -
always easy to prove ? - the plot thickens!


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