## prime, prime square, prime cube

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• Hello group, [subj] can form an arithmetic progression, e.g. 23, 25, 27 Are there any more examples known? Best regards, Andrey [Non-text portions of this
Message 1 of 9 , Feb 23, 2012
Hello group,

[subj] can form an arithmetic progression, e.g.

23, 25, 27

Are there any more examples known?

Best regards,

Andrey

[Non-text portions of this message have been removed]
• ... If 3 numbers in AP is your only condition then there are many small examples. Below are the 42 cases with all numbers below 10^6, sorted by the square and
Message 2 of 9 , Feb 23, 2012
Andrey Kulsha wrote:
> [subj] can form an arithmetic progression, e.g.
>
> 23, 25, 27
>
> Are there any more examples known?

If 3 numbers in AP is your only condition then there are many small examples.
Below are the 42 cases with all numbers below 10^6, sorted by the square
and then the cube.

23, 25=5^2, 27=3^3, common difference 2
541, 1369=37^2, 2197=13^3, common difference 828
103, 3481=59^2, 6859=19^3, common difference 3378
5623, 6241=79^2, 6859=19^3, common difference 618
2467, 16129=127^2, 29791=31^3, common difference 13662
19507, 24649=157^2, 29791=31^3, common difference 5142
14869, 32761=181^2, 50653=37^3, common difference 17892
28549, 39601=199^2, 50653=37^3, common difference 11052
58831, 69169=263^2, 79507=43^3, common difference 10338
73951, 76729=277^2, 79507=43^3, common difference 2778
157, 113569=337^2, 226981=61^3, common difference 113412
30781, 128881=359^2, 226981=61^3, common difference 98100
42397, 134689=367^2, 226981=61^3, common difference 92292
1879, 151321=389^2, 300763=67^3, common difference 149442
88237, 157609=397^2, 226981=61^3, common difference 69372
94621, 160801=401^2, 226981=61^3, common difference 66180
107581, 167281=409^2, 226981=61^3, common difference 59700
50359, 175561=419^2, 300763=67^3, common difference 125202
53719, 177241=421^2, 300763=67^3, common difference 123522
144541, 185761=431^2, 226981=61^3, common difference 41220
147997, 187489=433^2, 226981=61^3, common difference 39492
176221, 201601=449^2, 226981=61^3, common difference 25380
190717, 208849=457^2, 226981=61^3, common difference 18132
201757, 214369=463^2, 226981=61^3, common difference 12612
47161, 218089=467^2, 389017=73^3, common difference 170928
181399, 241081=491^2, 300763=67^3, common difference 59682
12979, 253009=503^2, 493039=79^3, common difference 240030
242119, 271441=521^2, 300763=67^3, common difference 29322
49843, 271441=521^2, 493039=79^3, common difference 221598
209401, 299209=547^2, 389017=73^3, common difference 89808
105379, 299209=547^2, 493039=79^3, common difference 193830
231481, 310249=557^2, 389017=73^3, common difference 78768
224563, 358801=599^2, 493039=79^3, common difference 134238
362521, 375769=613^2, 389017=73^3, common difference 13248
258499, 375769=613^2, 493039=79^3, common difference 117270
273283, 383161=619^2, 493039=79^3, common difference 109878
303283, 398161=631^2, 493039=79^3, common difference 94878
375523, 434281=659^2, 493039=79^3, common difference 58758
380803, 436921=661^2, 493039=79^3, common difference 56118
215329, 564001=751^2, 912673=97^3, common difference 348672
402769, 657721=811^2, 912673=97^3, common difference 254952
776449, 844561=919^2, 912673=97^3, common difference 68112

--
Jens Kruse Andersen
• Thank you Jens for the quick reply. ... Smaller common differences are of greater interest of course :-) Heuristically the number of examples is finite for a
Message 3 of 9 , Feb 23, 2012
Thank you Jens for the quick reply.

> If 3 numbers in AP is your only condition then there are many small
> examples.
> Below are the 42 cases with all numbers below 10^6, sorted by the square
> and then the cube.
>
> 23, 25=5^2, 27=3^3, common difference 2
> 541, 1369=37^2, 2197=13^3, common difference 828
> 103, 3481=59^2, 6859=19^3, common difference 3378
> 5623, 6241=79^2, 6859=19^3, common difference 618

Smaller common differences are of greater interest of course :-)

Heuristically the number of examples is finite for a given common
difference.

Note that the difference can't be +/-1 (mod 5).

Best regards,

Andrey
• ... THEOREM: The number of examples is finite for a given common difference. BECAUSE: the number of solutions (x,y) of x^2 - y^3 = k is finite for any fixed k,
Message 4 of 9 , Feb 23, 2012
> Heuristically the number of examples is finite for a given common difference.

THEOREM: The number of examples is finite for a given common difference.

BECAUSE:
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.
QED
• ... 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 5 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 6 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 7 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 8 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 9 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
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