## Re: prime, prime square, prime cube

Expand Messages
• ... The actual history is both more complex and more interesting than Warren s mistaken attribution. J.W.S. Cassels, Mordell s finite basis theorem revisited,
Message 1 of 9 , Feb 24, 2012
"WarrenS" <warren.wds@...> wrote:

> the number of solutions (x,y) of x^2 - y^3 = k
> is finite for any fixed k,
> proved by LJ Mordell sometime between 1905-1925.

The actual history is both more complex and more

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
• ... --David, thanks a lot for this help. I in fact was going to look into this myself but unfortunately my library only allowed me to access one of Mordell s 3
Message 2 of 9 , Feb 28, 2012
> The actual history is both more complex and more
> interesting than Warren's mistaken attribution.

--David, thanks a lot for this help.
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.
• ... --David, thanks a lot for this help. I in fact was going to look into this myself but unfortunately my library only allowed me to access one of Mordell s 3
Message 3 of 9 , Feb 28, 2012
> The actual history is both more complex and more
> interesting than Warren's mistaken attribution.

--David, thanks a lot for this help.
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.
• ... I not have read either of those. I have, however, read L. J. Mordell, On the Integer Solutions of the Equation, e*y^2 = a*x^3+b*x^2+c*x+d, Proc. London
Message 4 of 9 , Feb 28, 2012
"WarrenS" <warren.wds@...> wrote:

> > The actual history is both more complex and more
> > 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
finitely many integral solutions for any cubic form
(or more generally, for any other than a power of a
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
• ... Unless there just happens to be one or more of the form 5, x^2 = 5 + k, y^3 = 5 + 2k Kermit
Message 5 of 9 , Mar 4, 2012
> Note that the difference can't be ±1 (mod 5).

> Best regards,

> Andrey

Unless there just happens to be one or more of the form

5, x^2 = 5 + k, y^3 = 5 + 2k

Kermit
Your message has been successfully submitted and would be delivered to recipients shortly.