History of Congruent Numbers problem

[Geoff]

On page 2 of Introduction to Elliptic Curves and Modular Forms by
Koblitz, he says "Euler was the first to show that n = 7 is a
congruent number." I emailed him about this and he did not remember
the source. However, in Dickson's book History of Number Theory
Volume 2 P462, it says, "Leonardo [Fibonacci] noted that many numbers
are not congruent; but any number is a congruent if the quotient of
any congruent number by it is a square. A number is congruent if it
equals one of the four numbers a, b, a+b, a-b, and if the three are
squares. For example, 16, 9, 16+9 are squares, so that 16-9=7 is a
congruent number."

The problem is, Dickson's version is not clear. It seems to imply
that Fibonacci came up with this method but that's not even that
clear. Then, it gives the example but it certainly does not imply
that Fibonacci came up with it. It seems more of the author just
showing an easy example. And, if you read the previous pages to that,
it seems most of what is said about Fibonacci is about a slightly
different problem and 7 is not even a solution to that problem. For
example, at the bottom of P461 it says, "Thus he was able to state
that any congruent number is a multiple of 24." This is not true
about what is currently defined as a "congruent" number.

Does any one know anything more about this? I have seen people say
Fibonacci was the first to find that 7 is a congruent number, but my
guess is they are using this as the source, based on other things they
say that are also included in Dickson's book. And, I think they are
incorrect. But, I have never seen a source saying Euler was the first
either.

A separate question, I think I have seen one source that seemed to
indicate that Heegner was the first to relate the problem to elliptic
curves. Does any one know if this is true or not?

Thanks

[Gerry Myerson]

In article
<b88976f5-2b3c-4160...@m38g2000yqd.googlegroups.com>,
Geoff <geoffr...@gmail.com> wrote:

Here's what Guy says on p. 306 of Unsolved Problems in Number Theory,
3rd ed.:

Dickson's History gives many early references, including Leonardo
of Pisa (Fibonacci); Genocchi; and Gerardin, who gave 7, 22, 41, 69, 77,
the twenty Arabic examples and the forty-three entries CG in Table 7.

So perhaps Guy is attributing 7 to Gerardin. Guy gives the reference,
A. Gerardin, Nombres congruents, Intermediare Math. 22 (1915) 52-53.

> A separate question, I think I have seen one source that seemed to
> indicate that Heegner was the first to relate the problem to elliptic
> curves. Does any one know if this is true or not?

Perhaps some history is given in Tunnell's papers.

--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)

[GeoffTims]

I am replying to myself as I have found the answer to my own question, I believe, and I want to be sure that if any one else searches for this, they can see the answer as well.

The answer is that Fibonacci discovered 7 is a congruent number first and Euler also did after him, and this is perhaps why Koblitz attributes it to Euler. The answer is in Dickson, but a few pages past where I was looking, so I admit that I should have read farther originally, though I thought I had read the entire portion relating to Fibonacci and that would be enough. Dickson says in his history, Volume II:

"L Euler noted (as had Leonardo) that p^2 +- 5q^2 are both squares for p=41, q=12; p^2 +- 7q^2 both squares for p=337, q=120."