通分と分子計算が絶妙な関係をもつ組合せ
(1/n)=(1/a1)+(1/a2) の関係式を満たす組合せの調査
2=>
[3, 6]
*1/2=1/3+1/6の式が成り立つことを示す。
3=>
[4, 12]
4=>
[6, 12]
5=>
[6, 30]
6=>
[10, 15]
9=>
[12, 36]
10=>
[15, 30]
--------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)
2=>
[4, 6, 12]
3=>
[6, 10, 15]
4=>
[10, 12, 15]
5=>
[12, 15, 20]
6=>
[12, 21, 28]
7=>
[15, 21, 35]
9=>
[20, 30, 36]
[21, 28, 36]
10=>
[21, 35, 42]
------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)
2=>
[4, 10, 12, 15]
3=>
[9, 10, 15, 18]
4=>
[9, 18, 21, 28]
[10, 15, 21, 28]
5=>
[15, 20, 21, 28]
6=>
[20, 21, 28, 30]
7=>
[18, 28, 36, 42]
[20, 28, 30, 42]
9=>
[20, 35, 60, 63]
10=>
[30, 36, 45, 60]
-----------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)
2=>
[6, 9, 10, 15, 18]
3=>
[10, 12, 15, 21, 28]
4=>
[12, 20, 21, 28, 30]
5=>
[18, 21, 28, 30, 36]
6=>
[21, 28, 30, 36, 45]
7=>
[28, 30, 36, 42, 45]
9=>
[35, 36, 45, 60, 63]
10=>
[28, 45, 63, 70, 84]
[30, 42, 60, 70, 84]
[30, 45, 60, 63, 84]
[36, 42, 45, 70, 84]
--------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)
2=>
[5, 9, 18, 20, 21, 28]
[6, 9, 12, 18, 21, 28]
[6, 10, 12, 15, 21, 28]
[7, 9, 12, 14, 18, 28]
[7, 10, 12, 14, 15, 28]
3=>
[10, 15, 20, 21, 28, 30]
4=>
[18, 20, 21, 28, 30, 36]
5=>
[20, 21, 35, 36, 42, 45]
6=>
[21, 35, 36, 42, 45, 60]
7=>
[28, 35, 42, 45, 60, 63]
9=>
[35, 42, 60, 63, 70, 84]
10=>
[42, 45, 60, 70, 84, 90]
----------------------------------------------------
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)+(1/a7)
2=>
[9, 10, 12, 15, 18, 21, 28]
3=>
[14, 15, 20, 21, 28, 30, 35]
4=>
[15, 21, 30, 35, 36, 42, 45]
5=>
[21, 30, 35, 36, 42, 45, 60]
6=>
[28, 30, 35, 45, 60, 63, 70]
[30, 35, 36, 42, 45, 60, 70]
7=>
[30, 35, 45, 60, 63, 70, 84]
9=>
[42, 45, 60, 63, 84, 90, 105]
10=>
[42, 60, 63, 70, 84, 105, 126]
[45, 60, 63, 70, 84, 90, 126]
＊他にも多くの関係式が存在できますが最後に現れる数がなるだけ小さくなる
ものを選んで掲示しています。
-----------------------------------------------------------
平方数での関係式では
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2 の関係式を満たす組合せの調査
6=>
[7, 14, 21]
*(1/6)^2=(1/7)^2+(1/14)^2+(1/21)^2 が成立することを示す。
----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2
4=>
[5, 7, 28, 35]
----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2
4=>
[6, 7, 12, 14, 21]
6=>
[7, 15, 21, 42, 105]
9=>
[12, 14, 60, 252, 420]
10=>
[12, 21, 36, 252, 1260]
-----------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2+(1/a6)^2
3=>
[4, 6, 7, 60, 84, 420]
4=>
[6, 7, 14, 15, 20, 21]
5=>
[6, 10, 30, 35, 70, 105]
6=>
[7, 12, 60, 105, 140, 420]
[7, 15, 20, 60, 84, 420]
7=>
[12, 14, 15, 20, 28, 84]
9=>
[10, 30, 35, 70, 90, 105]
10=>
[12, 20, 60, 70, 140, 210]
------------------------------------------------------------
(1/n)^2=(1/a1)^2+(1/a2)^2+(1/a3)^2+(1/a4)^2+(1/a5)^2+(1/a6)^2+(1/a7)^2
3=>
[4, 6, 9, 12, 36, 45, 60]
[4, 6, 10, 12, 20, 30, 60]
4=>
[5, 10, 14, 15, 28, 30, 42]
5=>
[6, 12, 20, 21, 60, 84, 105]
6=>
[9, 12, 15, 20, 36, 45, 60]
7=>
[9, 14, 28, 36, 45, 60, 84]
[10, 14, 20, 28, 30, 60, 84]
9=>
[12, 20, 21, 60, 84, 90, 105]
10=>
[12, 28, 35, 42, 70, 84, 140]
[14, 20, 30, 35, 60, 84, 140]
----------------------------------------------------------
また立方数での関係式で調査してみました。
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3
10=>
[12, 15, 20]
*(1/10)^3=(1/12)^3+(1/15)^3+(1/20)^3 が成立することを示す。
----------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3
5=>
[6, 7, 15, 21, 30, 42, 210]
6=>
[7, 10, 14, 15, 30, 42, 70]
9=>
[10, 15, 30, 36, 45, 60, 90]
10=>
[12, 14, 30, 42, 60, 84, 420]
--------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3+(1/a8)^3
7=>
[9, 10, 14, 18, 63, 70, 105, 315]
9=>
[10, 14, 70, 84, 90, 105, 140, 210]
--------------------------------------------------------------------
(1/n)^3=(1/a1)^3+(1/a2)^3+(1/a3)^3+(1/a4)^3+(1/a5)^3+(1/a6)^3+(1/a7)^3+(1/a8)^3+(1/a9)^3
4=>
[5, 6, 7, 28, 35, 45, 252, 630, 1260]
5=>
[6, 7, 14, 30, 36, 42, 45, 60, 70]
6=>
[7, 10, 14, 15, 36, 42, 45, 60, 70]
9=>
[10, 15, 28, 36, 63, 70, 90, 180, 1260]
[10, 18, 20, 28, 36, 63, 70, 90, 1260]
10=>
[12, 14, 30, 42, 63, 84, 140, 180, 210]
などが構成可能になるようです。
(1/n)=(1/a1)+(1/a2) で
7⇒[8,56] とか 8⇒[9,72] はなぜ書かれていないのでしょう？
一般に n⇒[n+1,n(n+1)] ですね。
7⇒[8,56] とか 8⇒[9,72] が見逃された理由
N=2^a*3^b*5^c*7^d
(a=0,1,2;b=0,1,2;c=0,1;d=0,1)
なる因子に限定する３６タイプの数の組み合わせから、条件を満たす組合せを
探し出していたので、上記の数での組み合わせが顔を出さない結果となっていました。
ですから8=>に対するパターンがどの分野でも見逃される結果を招いています。
探す数の材料を
N=2^a*3^b*5^c＊7^d
(a=0,1,2,3;b=0,1,2;c=0,1;d=0,1)
48パターンでやってみました。
2=>
[3, 6]
3=>
[4, 12]
4=>
[5, 20]
[6, 12]
5=>
[6, 30]
6=>
[7, 42]
[8, 24]
[9, 18]
[10, 15]
7=>
[8, 56]
8=>
[9, 72]
[10, 40]
[12, 24]
9=>
[10, 90]
[12, 36]
10=>
[12, 60]
[14, 35]
[15, 30]
これでやっと姿が現れてきます。
(1/n)=(1/a1)+(1/a2)+(1/a3)+(1/a4)+(1/a5)+(1/a6)+(1/a7) で欠損している部分でも
8=>
[35, 42, 60, 63, 70, 72, 84]
[40, 42, 56, 60, 63, 72, 84]
その他多くが発見できました。
