- --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:

> the number of solutions (x,y) of x^2 - y^3 = k

The actual history is both more complex and more

> is finite for any fixed k,

> proved by LJ Mordell sometime between 1905-1925.

interesting than Warren's mistaken attribution.

J.W.S. Cassels,

Mordell's finite basis theorem revisited,

Math. Proc. Camb. Phil. Soc. 100 (1986) 31-40

points out that in the conclusion of

L. J. Mordell,

The diophantine equation y^2 k = x^3,

Proc. London Math. Soc. (2) 13 (1913) 60-80,

Mordell (not knowing of Thue's work) and was led,

for unknown reasons, to make a (false) claim directly

opposed to the result that Warren has attributed to him.

I looked to see if that is the case. Indeed it is.

Mordell concluded:

>> When k = 1, 4, 6, 7, 11, 18, 14, 16, 20, 21, 28, 25, 27,

29, 82, 34, 89, 42, 45, 46, 47, 49, 51, 58, 58, 59, 60, 61,

62, 66, 67, 69, 70, 75, 77, 78, 88, 84, 85, 86, 87, 88, 90,

98, 95, 96, the equations are insoluble or admit only a

limited number of solutions. For the remaining values of k,

there are an infinite number of solutions, except when

k = 74, in which case nothing can be said about the equation.<<

Mordell's 1913 paper gained him the Smith Prize,

but failed to gain him a Fellowship of St John's.

It seems that Mordell came to realize his error after studying

E. Landau and A. Ostrowski,

On the diophantine equation a*y^2 + b*y + c = d*x^n,

Proc. London Math. Soc. (2) 19 (1920) 276-280

and commented (in 1922) that this immediately

implies that he was wrong in 1913.

However, it appears that the paper by Landau and Ostrowski

appeared after the same result had been published in

A. Thue, Über die Unlösbarkeit der Gleichung

a*x^2 + b*x + c = d*y^n in grossen ganzen Zahlen,

Arch. math, og naturv. Kristiania 34 (1917) no. 16.

Of the 4 papers that I cite above,

I have read only the first two.

David >DJ Broadhurst:

--David, thanks a lot for this help.

> The actual history is both more complex and more

> interesting than Warren's mistaken attribution.

I in fact was going to look into this myself but unfortunately my library only allowed

me to access one of Mordell's 3 papers, namely the one you mentioned

PLMS 13 (1913) 60-80 which as you said (& I already knew) was not doing the job.

But I have reason to believe, but unconfirmed since I do not have access to these papers,

that one or both of the following papers by Mordell do do the job:

Cambridge Philos Soc Proc 21(1922) 179-192.

Quart J Pure Appl Math 45 (1914) 170-186.>DJ Broadhurst:

--David, thanks a lot for this help.

> The actual history is both more complex and more

> interesting than Warren's mistaken attribution.

I in fact was going to look into this myself but unfortunately my library only allowed

me to access one of Mordell's 3 papers, namely the one you mentioned

PLMS 13 (1913) 60-80 which as you said (& I already knew) was not doing the job.

But I have reason to believe, but unconfirmed since I do not have access to these papers,

that one or both of the following papers by Mordell do do the job:

Cambridge Philos Soc Proc 21(1922) 179-192.

Quart J Pure Appl Math 45 (1914) 170-186.- --- In primenumbers@yahoogroups.com,

"WarrenS" <warren.wds@...> wrote:

> > The actual history is both more complex and more

I not have read either of those. I have, however, read

> > interesting than Warren's mistaken attribution.

>

> I have reason to believe, but unconfirmed since I do not

> have access to these papers, that one or both of the

> following papers by Mordell do do the job:

> Cambridge Philos Soc Proc 21(1922) 179-192.

> Quart J Pure Appl Math 45 (1914) 170-186.

L. J. Mordell,

On the Integer Solutions of the Equation,

e*y^2 = a*x^3+b*x^2+c*x+d,

Proc. London Math. Soc. s2-21(1) (1923) 415-419

doi:10.1112/plms/s2-21.1.415

where Mordell concedes that this problem

was solved by Thue in 1909, with a result very

different from Mordell's false claim of 1913.

In Cassel's obituary of Mordell:

Louis Joel Mordell, 1888-1972,

by J. W. S. Cassels, in

Biographical Memoirs of Fellows of the Royal Society,

Vol. 19 (Dec., 1973), pp. 493-520.

there is mention of an earlier retraction, circa 1919-1920:

>> In a paper which was little noticed at the time Thue

had already shown that the equation (2.7) has only

finitely many integral solutions for any cubic form

(or more generally, for any other than a power of a

binary quadratic). If Mordell had known of this theorem,

he would have deduced at once that y^2 = x^3+ k has

only finitely many integral solutions x,y; at least with

x prime to 2k. As a matter of fact he only learned

of Thue's result later (paper 15) and at the time

he believed that there could be infinitely many solutions

for some k (cf. end of paper 2).<<

"Paper 15" seems to be:

L J Mordell,

A statement by Fermat,

Proc. Lond. Math. Soc., (2) 18 (1920) v.

The index for that volume indicates that this paper

was read in 1919, but not printed in the journal.

So 1919 is the earliest date for a retraction, by Mordell,

of his false claim of 1913, that I seen cited.

David