■2乗和が等しい数列とスー・モース数列(その61)

 1^0+13^0+28^0+70^0+82^0+124^0+139^0+151^0

=4^0+7^0+34^0+61^0+91^0+118^0+145^0+148^0  (等式)

 1^1+13^1+28^1+70^1+82^1+124^1+139^1+151^1

=4^1+7^1+34^1+61^1+91^1+118^1+145^1+148^1  (等式)

 1^2+13^2+28^2+70^2+82^2+124^2+139^2+151^2

=4^2+7^2+34^2+61^2+91^2+118^2+145^2+148^2  (等式)

 1^3+13^3+28^3+70^3+82^3+124^3+139^3+151^3

=4^3+7^3+34^3+61^3+91^3+118^3+145^3+148^3  (等式)

 1^4+13^4+28^4+70^4+82^4+124^4+139^4+151^4

=4^4+7^4+34^4+61^4+91^4+118^4+145^4+148^4  (等式)

 1^5+13^5+28^5+70^5+82^5+124^5+139^5+151^5

=4^5+7^5+34^5+61^5+91^5+118^5+145^5+148^5  (等式)

 1^6+13^6+28^6+70^6+82^6+124^6+139^6+151^6

=4^6+7^6+34^6+61^6+91^6+118^6+145^6+148^6  (等式)

 1^7+13^7+28^7+70^7+82^7+124^7+139^7+151^7

=4^7+7^7+34^7+61^7+91^7+118^7+145^7+148^7  (等式)

 1^8+13^8+28^8+70^8+82^8+124^8+139^8+151^8

≠4^8+7^8+34^8+61^8+91^8+118^8+145^8+148^8  (非等式)

 それにしても驚異的な例である.

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 {an}={1,4,6,7,10,11,13,16}

 {bn}={2,3,5,8,9,12,14,15}

と比較してみましょう。

 {an}={1,13,28,70,82,124,139,151}

 {bn}={4, 7,34,61,91,118,145,148}

 8対{1,4},・・・,{151,148}で考えると{an},{bn}にはそれぞれ4つの偶数,4つの奇数が属しています.

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{an^2}={1,169, 784,4900,6724,15376,19321,22801}

{bn^2}={16,49,1156,3721,8281,13924,21025,21904}

{cn}={bn^2−an^2}={15,−120,372,−1179,1557,−1452,1704,−897}=0

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1+151=152

13+139=152

28+124=152

70+82=152

4+148=152

7+145=152

34+118=152

61+91=152

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(76-75)^2+(76-63)^2+(76-48)^2+(76-6)^2+(76+6)^2+(76+48)^2+(76+63)^2+(76+75)^2

=(76-72)^2+(76-69)^2+(76-42)^2+(76-15)^2+(76+15)^2+(76+42)^2+(76+69)^2+(76+72)^2

より

75^2+63^2+48^2+6^2=72^2+69^2+42^2+15^2

25^2+21^2+16^2+2^2=24^2+23^2+14^2+5^2

625+441+256+4=576+529+196+25=1326

このほうがわかりやすい

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