[seqfan] Re: a Diophantine equation

[Max Alekseyev]

If m=2ab where a^2+b^2=c^2 then the number m/4=ab/2 is a congruent
number (an element of A003273) - see
http://en.wikipedia.org/wiki/Congruent_number for definition and basic
properties.
It is also clear that m/4 is congruent iff the square-free kernel
A007947(m) is such.

 > From this perspective, it is easy to see that A007947(7!) = 35 does
not belong to A003273, implying that 7! cannot be represented in the
form 2ab with a^2+b^2=c^2. Similarly, A007947(14!)=858 and
A007947(21!)=969969 are not congruent either.

However, the converse is not true - even if A007947(n!) is a congruent
number, the required integers a,b,c may not exist.
The smallest example is n=11 for which A007947(11!)=77 is congruent
but exhaustively search gives no integer solutions a,b,c.

An alternative view to the original question is to consider the three squares:
(a-b)^2, c^2, (a+b)^2
and notice that they form an arithmetic progression with the
difference equal n! (i.e., n! belongs to A057102). There is a somewhat
related study of arithmetic progressions of three rational squares in:
http://www.math.uconn.edu/~kconrad/ross2007/3squarearithprog.pdf

I also confirm that Jack Brennen's solutions are complete for n up to 22.
The following addition makes the solutions complete for n up to 30:

23! = 2 * 83545862400 * 154717516800
24! no solutions
25! no solutions
26! = 2 * 134316040320000 * 1501278105600
= 2 * 12817284480000 * 15732328550400
= 2 * 16354735488000 * 12329501184000
= 2 * 1117404288000 * 180459062784000
27! = 2 * 42237882086400 * 128899330560000
28! no solutions
29! no solutions
30! no solutions

Regards,
Max

On Tue, Nov 11, 2008 at 6:09 PM, David Newman <davidsnewman at gmail.com> wrote:
> A student,  named Daniel Reichman, asked this question.
>
> Suppose that a,b, and c, integers, satisfy
>
> a^2+b^2= c^2.
>
> For which n are there a,b such that 2ab=n!  ?
>
> He notes that there are at least three solutions:
>
> For n=4, a=4, b=3
> For n=5  a=12, b=5
> For n=6, a=40, b=9
>
> Are there any other such n?
>
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