where a, b, c are integers

Paul found:

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where a, b, c are integers

Paul found:

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%d bloggers like this:

Here are a few more

104^4 + 239^4 = 58136^2 + 1

111^4 + 152^4 = 26184^2 + 1

113^4 + 319^4 = 102559^2 + 1

118^4 + 705^4 = 497220^2 + 1

143^4 + 239^4 = 60671^2 + 1

157^4 + 253^4 = 68591^2 + 1

193^4 + 668^4 = 447776^2 + 1

208^4 + 239^4 = 71656^2 + 1

257^4 + 560^4 = 320480^2 + 1

295^4 + 485^4 = 250807^2 + 1

305^4 + 643^4 = 423785^2 + 1

319^4 + 794^4 = 638596^2 + 1

401^4 + 664^4 = 469304^2 + 1

505^4 + 515^4 = 367943^2 + 1

560^4 + 785^4 = 691432^2 + 1

577^4 + 624^4 = 512304^2 + 1

593^4 + 812^4 = 747256^2 + 1

643^4 + 905^4 = 917465^2 + 1

Paul.

Nice!

Ben, can I ask you a question.

Fermat, in his days, challenged his colleagues pretty often.

Like this one:

Find a cube n such that n + S(n) is a square, and

Find a square m such that m + S(m) is a cube.

Where S(n) is the sum of the proper divisors, or sigma(n) – n.

For question 1, Frenicle de Bessy found four solutions within one day and six other solutions the next day.

One is: 7^3. S(7^3) = 1 + 7 + 7^2 = 57, so 7^3 + 57 = 400, clearly a square. Not that hard to find but the only answer I am able to find.

For the second question I can not find any answers.

My question to you is: can you find the other nine solutions for question 1, and answers for question 2?

And how where these guys able to find big solutions in the 17th century?

pipo

n, sum of divisors:

170, 324 = 18^2

4097, 4356 = 66^2

4490, 8100 = 90^2

35722, 54756 = 234^2

118337, 125316 = 354^2

810001, 813604 = 902^2

6245002, 9400356 = 3066^2

9168785, 11696400 = 3420^2

18088010, 32558436 = 5706^2

Sorry, I misread your question.

These are all solutions for a cube equal to a square Pt 1.

{7,751530,4730879,5260710,33116153,37200735,187062910,226141311,259109835,260405145,370049418,522409465,836308083,1105725765,1309440370,1343713507,1582989177,1609505430,1813768845,2590345926,3039492538,3656866255,5854156581,7109214762,7740080355,9405994549,11266538010,20145398286,21276447766,22889024957,24839708520,44584153947,49316447830,49764503334,51303747330,59480154024,78339618370,79670518785,86415819433}

and for the square equals a cube Pt 2.

{43098,20746664124,21531558370,25933330155,30519275171,453393100534,803844998180,1233758294601,2358796315843,6260406046762,7339897643091,7540242750903,8532869860592,11879890160946,17538398093508}

Paul.

Didn’t come out too well so here are those in column format

7

751530

4730879

5260710

33116153

37200735

187062910

226141311

259109835

260405145

370049418

522409465

836308083

1105725765

1309440370

1343713507

1582989177

1609505430

1813768845

2590345926

3039492538

3656866255

5854156581

7109214762

7740080355

9405994549

11266538010

20145398286

21276447766

22889024957

24839708520

44584153947

49316447830

49764503334

51303747330

59480154024

78339618370

79670518785

86415819433

and

43098

20746664124

21531558370

25933330155

30519275171

453393100534

803844998180

1233758294601

2358796315843

6260406046762

7339897643091

7540242750903

8532869860592

11879890160946

17538398093508

Paul.

Wow, thanks Paul