a^4=0,1,16,81,256,・・・

1,2,17,82,

16,17,32,97

81,82,97

a^4+b^4=0,1,2,16,17,32,81,82,97,・・・

これに対しては15定理は成立しない。

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mod8で見ると

a^4=0,1,0,1,・・・

a^4+b^4=0,1,2

a^4+b^4+c^4=0,1,2,3

a^4+b^4+c^4+d^4=0,1,2,3,4

大きな数はこれでいいのであるが、小さな数が8つの四乗数の和で表されるとは限らない

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a^4+b^4=0,1,2,16,17,32,81,82,97,・・・

0,1,2,16,17,32,81,82,97,・・・

1,2,3,17,18,33,82,83,98,・・・

2,3,4,18,19,34,83,84,99,・・・

16,17,18,32,33,48,97,98,・・・

17,18,19,33,34,49,98,99,・・・

32,33,34,48,49,64,・・・

81,82,83,97,98,・・・

82,83,84,98,99,・・・

97,98,99,・・・

a^4+b^4+c^4=0,1,2,3,4,16,17,18,19,32,33,34,48,49,81,82,83,84,97,98,99,・・・

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0,1,2,3,4,16,17,18,19,32,33,34,48,49,81,82,83,84,97,98,99,・・・

1,2,3,4,5,17,18,19,20,33,34,35,49,50,82,83,84,85,98,99,100,・・・

2,3,4,5,6,18,19,20,21,34,35,36,50,51,83,84,85,86,98,99,100,・・・

3,4,5,6,7,19,20,21,22,35,36,37,51,52,84,85,86,87,100,・・・

4,5,6,7,8,20,21,22,23,36,37,38,52,53,85,86,87,88,・・・

16,17,18,19,20,32,33,34,35,48,49,50,64,65,97,98,99,100,・・・

17,18,19,20,21,33,34,35,36,49,50,51,65,66,98,99,100,・・・

18,19,20,21,22,34,35,36,37,50,51,52,66,67,99,100,・・・

19,20,21,22,23,35,36,37,38,51,52,53,67,68,100,・・・

32,33,34,35,36,48,49,50,51,64,65,66,80,81,・・・

33,34,35,36,37,49,50,51,52,65,66,67,81,82,・・・

34,35,36,37,38,50,51,52,53,66,67,68,82,83,・・・

48,49,50,51,52,64,65,66,67,80,81,82,96,97,・・・

49,50,51,52,53,65,66,67,68,81,82,83,97,98,・・・

81,82,83,84,85,97,98,99,100,・・・

82,83,84,85,86,98,99,100,・・・

83,84,85,86,87,99,100,・・・

84,85,86,87,88,100,・・・

97,98,99,100,・・

a^4+b^4+c^4+d^4=

0,1,2,3,4,5,6,7,8,16,17,18,19,20,21,22,23,32,33,34,35,36,37,38,48,49,50,51,52,53,64,65,66,67,68,80,81,82,83,84,85,86,87,88,96,97,98,99,100,・・・

やはり79が最後になりそうである

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