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

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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

  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

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