## Equation: a^4 + b^4 = c^2 + 1

$a^4 \; + \; b^4 \; = \; c^2 \; + \; 1$

where   a, b, c   are integers

$5^4 \; + \; 7^4 \; = \; 55^2 \; + \; 1$
$8^4 \; + \; 17^4 \; = \; 296^2 \; + \; 1$
$13^4 \; + \; 13^4 \; = \; 239^2 \; + \; 1$
$17^4 \; + \; 32^4 \; = \; 1064^2 \; + \; 1$
$22^4 \; + \; 31^4 \; = \; 1076^2 \; + \; 1$
$27^4 \; + \; 37^4 \; = \; 1551^2 \; + \; 1$
$28^4 \; + \; 47^4 \; = \; 2344^2 \; + \; 1$
$31^4 \; + \; 46^4 \; = \; 2324^2 \; + \; 1$
$44^4 \; + \; 63^4 \; = \; 4416^2 \; + \; 1$
$46^4 \; + \; 97^4 \; = \; 9644^2 \; + \; 1$
$47^4 \; + \; 76^4 \; = \; 6184^2 \; + \; 1$
$64^4 \; + \; 111^4 \; = \; 12984^2 \; + \; 1$
$91^4 \; + \; 99^4 \; = \; 12831^2 \; + \; 1$
$98^4 \; + \; 191^4 \; = \; 37724^2 \; + \; 1$

Paul found:

$104^4 + 239^4 = 58136^2 + 1$
$111^4 + 152^4 = 26184^2 + 1$
$113^4 + 319^4 = 102559^2 + 1$
$118^4 + 705^4 = 497220^2 + 1$
$143^4 + 239^4 = 60671^2 + 1$
$157^4 + 253^4 = 68591^2 + 1$
$193^4 + 668^4 = 447776^2 + 1$
$208^4 + 239^4 = 71656^2 + 1$
$257^4 + 560^4 = 320480^2 + 1$
$295^4 + 485^4 = 250807^2 + 1$
$305^4 + 643^4 = 423785^2 + 1$
$319^4 + 794^4 = 638596^2 + 1$
$401^4 + 664^4 = 469304^2 + 1$
$505^4 + 515^4 = 367943^2 + 1$
$560^4 + 785^4 = 691432^2 + 1$
$577^4 + 624^4 = 512304^2 + 1$
$593^4 + 812^4 = 747256^2 + 1$
$643^4 + 905^4 = 917465^2 + 1$

math grad - Interest: Number theory
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### 8 Responses to Equation: a^4 + b^4 = c^2 + 1

1. paul says:

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.

2. pipo says:

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

• benvitalis says:

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

• benvitalis says:

3. paul says:

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.

4. paul says:

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.

5. pipo says:

Wow, thanks Paul