[SeqFan] solution for Diophantine equation x^3+y^3=A.z^3 with A=284

[Patrick De Geest]

In https://oeis.org/A190356, https://oeis.org/A190580 and https://oeis.org/A190581

I found in the links sections the webpage https://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt

from Hisanori Mishima, about solutions of the Diophantine equation x^3+y^3=A.z^3

for the first 1000 integers A if known.

As I am building a website (in Dutch) about the properties of numbers from 0 to 1000 I also added

these solutions from the file x3y3s.txt. In this case https://www.worldofnumbers.com/AG0284.htm?box=r8

Yet with number 284 I ran into the following problem

x = 772263046200089644994113

y = -1293813622621939303367

z = 117487719436278023459434

as (x/z)^3+(y/z)^3. gives me using pari (note: y is negative)

283.99999866585440828218412381737102179...

which is not 284 but a fairly good approximation.

This leads me into thinking that perhaps somewhere in the values of x,y and/or z a digit is wrong (typo?), or

a digit(s) is missing out somewhere (or one too many). Does anyone know an alternative list of solutions that go up to 1000 ?

Maybe someone from this list is smart enough to recalculate or reprogram it altogether... or can find out

where the wrongdoing digit is hidden in x,y or z.  Thanks in advance for the effort to try to remedy this situation.

I tried to contact Mr Mishima at (kc2h...@asahi-net.or.jp) but alas to no avail.

-Patrick De Geest.

[Arthur O'Dwyer]

These numbers are small enough you can just use exact integer math (in, e.g., Python) instead of messing around with floating-point division.

>>> 772263046200089644994113**3 - 1293813622621939303367**3

460570120083650260395392055628867283631897988976273037508188175886869034

>>> 284 * 117487719436278023459434**3

460570122247268564174014987072973887831448410008850558214582523404215136

Or again

>>> x = 772263046200089644994113; y = -1293813622621939303367; z = 117487719436278023459434

>>> x**3 + y**3 == 284 * z**3

False

So indeed (x = 772263046200089644994113, y = -1293813622621939303367, z = 117487719436278023459434) is NOT a solution to x**3 + y**3 = 284 * z**3.

But from your email I can't tell why you thought it might be. Is there an actual mistake in any of those three OEIS sequences you listed?

HTH,

Arthur

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[David desJardins]

On Wed, Jul 23, 2025 at 5:20 PM Arthur O'Dwyer <arthur....@gmail.com> wrote:

So indeed (x = 772263046200089644994113, y = -1293813622621939303367, z = 117487719436278023459434) is NOT a solution to x**3 + y**3 = 284 * z**3.

But from your email I can't tell why you thought it might be. Is there an actual mistake in any of those three OEIS sequences you listed?

Because it's copied from the link he gave.

https://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt

[Daniel Mondot]

I confirm.

Even if you don't have the ability to multiply very large integers, to do a quick test, you can just do it mod 1000000, for instance (i.e. just keep the last 6 digits at every step)

so  x^3 + y^3 - 284 * z^3 mod 1000000 should be zero.

and 994113^3 - 303367^3 - 284*459434^3 = -346102 or 653898 (last 6 digits are not zero, so the equation is false.) 

I suspected that there must be a small typo in the last digits of one of the numbers, but haven't been able to find numbers near these that would make the equation true.

I now suspect that the author just didn't display enough decimals for A, and took these as a solution.

Daniel

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[Victor Miller]

There certainly should be a cross reference to 

https://oeis.org/A060838

.

It looks like this is asking for the smallest height non torsion point of the curve x^3+y^3=n.

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[Victor Miller]

From the table at 

https://www.kaynet.or.jp/~kay/misc/nna3.html

The solution for 284 is given as

195812865727130039099057283 : -3861962137811759194622252285 : 377608404165604191639171]

On Wed, Jul 23, 2025 at 09:12 Patrick De Geest <patrick.pa...@gmail.com> wrote:

[Victor Miller]

I looked at the wrong column. The solution should be

[-1293813622621939303367981 : 7722630462000896449941136589 : 1174877194362780234594343698

]

On Wed, Jul 23, 2025 at 09:12 Patrick De Geest <patrick.pa...@gmail.com> wrote:

[Arthur O'Dwyer]

Oh, I see now. Patrick De Geest was reading the table located at

https://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/x3y3s.txt

which purports to contain solutions to x**3 + y**3 == n * z**3. But it doesn't. These entries are all wrong:

WRONG:  284 772263046200089644994113 -1293813622621939303367 117487719436278023459434

WRONG:  356 150266304920614760419470 -4709632110011335573393 209822114158068144655458

WRONG:  501 656083776822609815827039 -6340671464215154829954 379984481300577822189297

WRONG:  781 452298962687879337114671 -3189898010858482837836 42527431999986819491970

Then, Victor Miller gave a link to a table located at

https://www.kaynet.or.jp/~kay/misc/nna3.html

which also purports to contain solutions to x**3 + y**3 == n * z**3. Python confirms that all of the entries in this latter table are correct.

The entries from https://www.kaynet.or.jp/~kay/misc/nna3.html which correct the wrong entries above are:

RIGHT:  284 7722630462000896449941136589 -1293813622621939303367981 1174877194362780234594343698

RIGHT:  356 15026630492061476041947013 -4709632110011335573393177 2098221141580681446554589

RIGHT:  501 65608377682260981582703907 -63406714642151548299544907 3799844813005778221892970

RIGHT:  781 452298962687879337114671 -318989801085848283783671 42527431999986819491970

However, I observe that the next page in that collection of tables — https://www.kaynet.or.jp/~kay/misc/nna4.html —contains many wrong entries too.

The wrong entries on https://www.kaynet.or.jp/~kay/misc/nna4.html are:

WRONG:  570 44391008927477 -23911621134284 5056073480199

WRONG:  642 18147157 12227843 2298150

WRONG:  662 3566251148564900 2140498649652379 437001508520199

WRONG:  673 866 646 111

WRONG:  703 9 -3 1

However, I notice that on https://www.kaynet.or.jp/~kay/misc/nna3.html and https://www.kaynet.or.jp/~kay/misc/nna4.html some entries have unexplained HTML formatting — red text, highlighting, etc. — which might be meant to indicate that the results are uncertain or wrong or something. The body text never explains what that formatting means.

–Arthur

[Patrick De Geest]

Thanks all for the replies re. this topic.

Especially to Mr Victor Miller who provided me with an alternative list of solutions

from Hisayasu Nakao also for A=284 through seq. A060838.

As end-digits where chopped of in the x,y & z solution from Mishima I guess it was a result of

an unfortunate copy and paste at the time.

With Nakao's list I can now check the correctness against that of Mishima's.

Thx also for Arthur O'Dwyer warning me of errors higher up in the lists of both sources.

These recent corrections are also very helpful.

In the near future I hope to add the crossreference (A060838) to various OEIS sequences together

with the links to H. Nakao lists.

With regards,

Patrick De Geest

Op donderdag 24 juli 2025 om 04:53:23 UTC+2 schreef victor...@gmail.com: