a2+b2+c2=d2 を満たす自然数a,b,c,dの組が無限に存在する。
ならば、何らかの一般式が存在するのではなかろうか。
ピタゴラス数を研究してきた私も、左辺が3項のものは手つかずにいました。
というわけで、プログラミングでローラー作戦をします。
a, b, cを1から100までループでネストして、自然数dが得られれば、それをデータとします。
なお、既約ピタゴラス数にしか興味がないので、gcd(a,b,c,d)=1となるものだけをデータとします。
すると、530組が見つかりました。
530組のデータの特徴をみてみると、
a,b,cのうち、奇数が1個、偶数が2個、dは奇数
という特徴がありました。
a,b,c,dがすべて奇数の可能性もありそうですが、1≦a,b,c≦100においては、a,b,cのいずれか1つが奇数、残り2つが偶数、dが奇数という組しか出てきませんでした。
この証明を考えないとダメですね。
まだまだ思考中です。
というわけで、まだまだ謎は残っていますが、
a,dを奇数、b,cを偶数、b≦cと固定して、(a,b,c,d)の様に空白なしでカンマ区切りで表示することとします。
これで、このページを検索すれば、一般式が存在するのか、存在するならばどのようなパラメータを与えたのかが解るかと思います。
続いて、一般式の可能性の有るものをいくつか揃えると、
【パターン1】
l,m,n∈N
gcd(l,m,n)=1
a=m2+n2-l2
b=2nl
b=2ml
c=m2+n2+l2
として、
b≦cを保つために、m≧n
a≧1を保つために、l2＜m2+n2
という条件を付加しました。
【パターン2】
l,m,n∈N
a=(2n-1)(2l-1)
b=(2m)(2l-1)
c=2(m2+n2-l2-n+l)
d=2(m2+n2+l2-n-l)+1
として、
b≦cを保つために、
b=min((2m)(2l-1),2(m2+n2-l2-n+l))
c=max((2m)(2l-1),2(m2+n2-l2-n+l))
のように変更しました。
gcd(l,m,n)=1を条件に加えていません。
なぜなら、
(l,m,n)=(2,2,2)のとき、(a,b,c,d)=(9,8,12,17)を得られます。
gcd(l,m,n)=2ですが、gcd(a,b,c,d)=1となります。
【パターン3】
m,n,p,q∈Z
m,p≧1, n,q≧0
gcd(m,n,o,p)=1
m+n+p+q≡1 (mod 2)
a=(m2+n2)-(p2+q2)
b=2(mp-nq)
c=2(mq+np)
d=(m2+n2)+(p2+q2)
として、
b≦cを保つために、
b=min(2(mp-nq),2(mq+np))
c=max(2(mp-nq),2(mq+np))
のように変更しました。
gcd(m,n,p,q)=1を条件に加えてありますが、
(m,n,p,q)=(1,7,5,0)のとき、(a,b,c,d)=(25,10,70,75)となり、
gcd(a,b,c,d)=5となってしまいます。
しかし、(a,b,c,d)/5=(25,10,70,75)/5=(5,2,14,15)が出ないわけではなく、
(m,n,p,q)=(3,1,1,2)のときに(a,b,c,d)=(5,2,14,15)は得られます。
gcd(m,n,o,p)=1とm+n+o+p≡1 (mod 2)だけでは、完全にgcd(a,b,c,d)=1とすることは出来ていません。
何かしらの条件が解れば追記します。
さて、3つのパターンがありますが、1≦a,b,c≦100というレンジで、どれくらいをカバー出来ているのでしょうか。
表にまとめると、
| 1≦a,b,c≦100 (a,b,c,d) | パターン1 (l,m,n) | パターン2 (l,m,n) | パターン3 (m,n,p,q) |
| (1,2,2,3) | (1,1,1) | (1,1,1) | (1,1,1,0) |
| (1,4,8,9) | (2,2,1) | (1,2,1) | (1,2,2,0) |
| (1,6,18,19) | (3,3,1) | (1,3,1) | (1,3,3,0) |
| (1,8,32,33) | (4,4,1) | (1,4,1) | (1,4,4,0) |
| (1,10,50,51) | (5,5,1) | (1,5,1) | (1,5,5,0) |
| (1,12,12,17) | (3,0,2,2) | ||
| (1,12,72,73) | (6,6,1) | (1,6,1) | (1,6,6,0) |
| (1,14,98,99) | (7,7,1) | (1,7,1) | (1,7,7,0) |
| (1,18,30,35) | (3,3,4,1) | ||
| (1,22,46,51) | (5,1,3,4) | ||
| (1,28,76,81) | (4,5,6,2) | ||
| (1,32,100,105) | (7,2,4,6) | ||
| (1,34,38,51) | (5,1,4,3) | ||
| (1,44,68,81) | (5,4,6,2) | ||
| (1,68,80,105) | (7,2,6,4) | ||
| (1,70,70,99) | (7,5,5) | (5,5,7,0) | |
| (3,2,6,7) | (1,1,2) | (2,1,1,1) | |
| (3,4,12,13) | (1,2,2) | (2,2,2,1) | |
| (3,6,22,23) | (1,3,2) | (2,3,3,1) | |
| (3,8,36,37) | (1,4,2) | (2,4,4,1) | |
| (3,10,54,55) | (1,5,2) | (2,5,5,1) | |
| (3,12,76,77) | (1,6,2) | (2,6,6,1) | |
| (3,14,18,23) | (2,3,1) | (3,2,3,1) | |
| (3,16,24,29) | (4,0,2,3) | ||
| (3,24,28,37) | (2,4,1) | (4,2,4,1) | |
| (3,24,56,61) | (4,4,5,2) | ||
| (3,26,66,71) | (6,1,3,5) | ||
| (3,30,46,55) | (2,5,1) | (5,2,5,1) | |
| (3,36,68,77) | (2,6,1) | (6,2,6,1) | |
| (3,42,94,103) | (2,7,1) | (7,2,7,1) | |
| (3,46,54,71) | (6,1,5,3) | ||
| (3,50,90,103) | (7,2,5,5) | ||
| (3,56,84,101) | (7,6,4) | (4,6,7,0) | |
| (3,80,96,125) | (8,0,5,6) | ||
| (3,92,96,133) | (8,2,7,4) | ||
| (5,2,14,15) | (1,1,3) | (3,1,1,2) | |
| (5,4,20,21) | (1,2,3) | (3,2,2,2) | |
| (5,6,30,31) | (1,3,3) | (3,3,3,2) | |
| (5,8,44,45) | (1,4,3) | (3,4,4,2) | |
| (5,10,62,63) | (1,5,3) | (3,5,5,2) | |
| (5,12,84,85) | (1,6,3) | (3,6,6,2) | |
| (5,38,50,63) | (3,5,1) | (5,3,5,2) | |
| (5,40,56,69) | (6,1,4,4) | ||
| (5,48,60,77) | (6,5,4) | (4,5,6,0) | |
| (5,54,78,95) | (7,1,6,3) | ||
| (5,70,86,111) | (3,7,1) | (7,3,7,2) | |
| (7,2,26,27) | (1,1,4) | (4,1,1,3) | |
| (7,4,4,9) | (1,2,2) | (2,2,1,0) | |
| (7,4,32,33) | (1,2,4) | (4,2,2,3) | |
| (7,6,6,11) | (3,0,1,1) | ||
| (7,6,42,43) | (1,3,4) | (4,3,3,3) | |
| (7,8,56,57) | (1,4,4) | (4,4,4,3) | |
| (7,10,74,75) | (1,5,4) | (4,5,5,3) | |
| (7,12,96,97) | (1,6,4) | (4,6,6,3) | |
| (7,14,22,27) | (4,1,3,1) | ||
| (7,16,28,33) | (4,2,3,2) | ||
| (7,24,60,65) | (6,0,2,5) | ||
| (7,26,70,75) | (4,5,1) | (5,4,5,3) | |
| (7,30,30,43) | (5,0,3,3) | ||
| (7,40,40,57) | (5,4,4) | (4,4,5,0) | |
| (7,48,84,97) | (4,6,1) | (6,4,6,3) | |
| (7,74,98,123) | (4,7,1) | (7,4,7,3) | |
| (9,2,6,11) | (1,3,1) | (2,1,2) | (1,3,1,0) |
| (9,2,42,43) | (1,1,5) | (5,1,1,4) | |
| (9,4,48,49) | (1,2,5) | (5,2,2,4) | |
| (9,6,58,59) | (1,3,5) | (5,3,3,4) | |
| (9,8,12,17) | (2,3,2) | (2,2,2) | (2,3,2,0) |
| (9,8,72,73) | (1,4,5) | (5,4,4,4) | |
| (9,10,90,91) | (1,5,5) | (5,5,5,4) | |
| (9,12,20,25) | (4,1,2,2) | ||
| (9,18,38,43) | (5,1,4,1) | ||
| (9,22,54,59) | (5,3,4,3) | ||
| (9,24,32,41) | (4,4,3) | (2,4,2) | (3,4,4,0) |
| (9,28,84,89) | (7,0,2,6) | ||
| (9,30,50,59) | (5,5,3) | (2,5,2) | (3,5,5,0) |
| (9,32,36,49) | (5,2,4,2) | ||
| (9,42,98,107) | (7,7,3) | (2,7,2) | (3,7,7,0) |
| (9,46,78,91) | (7,1,4,5) | ||
| (9,62,66,91) | (7,1,5,4) | ||
| (11,2,10,15) | (3,2,1,1) | ||
| (11,2,62,63) | (1,1,6) | (6,1,1,5) | |
| (11,4,68,69) | (1,2,6) | (6,2,2,5) | |
| (11,6,78,79) | (1,3,6) | (6,3,3,5) | |
| (11,8,16,21) | (4,0,1,2) | ||
| (11,8,92,93) | (1,4,6) | (6,4,4,5) | |
| (11,12,24,29) | (3,4,2) | (2,4,3,0) | |
| (11,18,42,47) | (5,2,3,3) | ||
| (11,22,58,63) | (6,1,5,1) | ||
| (11,28,88,93) | (6,4,5,4) | ||
| (11,36,48,61) | (6,0,3,4) | ||
| (11,42,66,79) | (6,3,5,3) | ||
| (11,44,52,69) | (6,2,5,2) | ||
| (11,58,94,111) | (5,6,7,1) | ||
| (11,74,82,111) | (6,5,7,1) | ||
| (13,2,86,87) | (1,1,7) | (7,1,1,6) | |
| (13,4,16,21) | (2,4,1) | (1,4,2,0) | |
| (13,4,92,93) | (1,2,7) | (7,2,2,6) | |
| (13,6,18,23) | (3,3,2,1) | ||
| (13,14,34,39) | (5,1,2,3) | ||
| (13,16,40,45) | (4,5,2) | (2,5,4,0) | |
| (13,24,72,77) | (6,3,4,4) | ||
| (13,26,82,87) | (7,1,6,1) | ||
| (13,50,70,87) | (5,5,6,1) | ||
| (13,52,76,93) | (7,2,6,2) | ||
| (13,66,78,103) | (7,3,6,3) | ||
| (13,76,88,117) | (8,1,6,4) | ||
| (15,6,10,19) | (2,1,3) | (4,1,1,1) | |
| (15,12,16,25) | (2,2,3) | (4,2,2,1) | |
| (15,18,26,35) | (2,3,3) | (4,3,3,1) | |
| (15,24,40,49) | (2,4,3) | (4,4,4,1) | |
| (15,30,58,67) | (2,5,3) | (4,5,5,1) | |
| (15,36,52,65) | (6,2,4,3) | ||
| (15,36,80,89) | (2,6,3) | (4,6,6,1) | |
| (15,42,50,67) | (3,5,2) | (5,4,5,1) | |
| (15,42,70,83) | (7,0,3,5) | ||
| (15,60,64,89) | (3,6,2) | (6,4,6,1) | |
| (15,90,94,131) | (8,3,7,3) | ||
| (17,4,28,33) | (4,3,2,2) | ||
| (17,6,6,19) | (1,3,3) | (3,3,1,0) | |
| (17,6,30,35) | (3,5,1) | (1,5,3,0) | |
| (17,14,46,51) | (3,5,4,1) | ||
| (17,16,52,57) | (6,1,2,4) | ||
| (17,20,20,33) | (5,0,2,2) | ||
| (17,24,84,89) | (6,7,2) | (2,7,6,0) | |
| (17,26,94,99) | (7,3,4,5) | ||
| (17,32,44,57) | (6,1,4,2) | ||
| (17,46,86,99) | (7,3,5,4) | ||
| (17,56,56,81) | (7,0,4,4) | ||
| (19,2,34,39) | (2,5,3,1) | ||
| (19,4,8,21) | (1,4,2) | (2,4,1,0) | |
| (19,8,40,45) | (4,4,3,2) | ||
| (19,12,48,53) | (6,0,1,4) | ||
| (19,18,66,71) | (3,6,5,1) | ||
| (19,22,26,39) | (5,2,3,1) | ||
| (19,22,82,87) | (7,2,3,5) | ||
| (19,30,42,55) | (6,1,3,3) | ||
| (19,42,54,71) | (6,3,5,1) | ||
| (19,48,96,109) | (8,0,3,6) | ||
| (19,58,62,87) | (7,2,5,3) | ||
| (21,2,42,47) | (5,3,2,3) | ||
| (21,6,22,31) | (2,1,4) | (5,1,1,2) | |
| (21,8,48,53) | (4,6,1) | (1,6,4,0) | |
| (21,12,16,29) | (2,4,3) | (3,4,2,0) | |
| (21,12,28,37) | (2,2,4) | (5,2,2,2) | |
| (21,12,56,61) | (4,4,2) | (4,5,4,2) | |
| (21,14,18,31) | (5,1,2,1) | ||
| (21,18,38,47) | (2,3,4) | (5,3,3,2) | |
| (21,18,74,79) | (7,1,2,5) | ||
| (21,22,90,95) | (3,7,6,1) | ||
| (21,24,52,61) | (2,4,4) | (5,4,4,2) | |
| (21,30,70,79) | (2,5,4) | (5,5,5,2) | |
| (21,36,92,101) | (2,6,4) | (5,6,6,2) | |
| (21,38,66,79) | (7,1,5,2) | ||
| (21,40,72,85) | (7,2,4,4) | ||
| (21,50,78,95) | (7,3,6,1) | ||
| (21,52,84,101) | (4,6,2) | (6,5,6,2) | |
| (21,72,100,125) | (8,3,6,4) | ||
| (21,78,98,127) | (4,7,2) | (7,5,7,2) | |
| (23,2,14,27) | (4,3,1,1) | ||
| (23,4,52,57) | (2,6,4,1) | ||
| (23,6,54,59) | (5,4,3,3) | ||
| (23,10,10,27) | (5,0,1,1) | ||
| (23,14,70,75) | (7,0,1,5) | ||
| (23,16,76,81) | (4,6,5,2) | ||
| (23,24,24,41) | (3,4,4) | (4,4,3,0) | |
| (23,24,36,49) | (6,0,2,3) | ||
| (23,28,44,57) | (6,2,4,1) | ||
| (23,44,64,81) | (6,4,5,2) | ||
| (23,64,80,105) | (8,0,4,5) | ||
| (23,72,84,113) | (8,2,6,3) | ||
| (23,84,84,121) | (7,6,6) | (6,6,7,0) | |
| (25,2,10,27) | (1,5,1) | (3,1,3) | (1,5,1,0) |
| (25,8,20,33) | (2,5,2) | (3,2,3) | (2,5,2,0) |
| (25,18,30,43) | (3,5,3) | (3,3,3) | (3,5,3,0) |
| (25,32,40,57) | (4,5,4) | (3,4,3) | (4,5,4,0) |
| (25,34,62,75) | (7,1,3,4) | ||
| (25,36,48,65) | (6,3,4,2) | ||
| (25,44,92,105) | (8,1,6,2) | ||
| (25,60,72,97) | (6,6,5) | (3,6,3) | (5,6,6,0) |
| (25,68,76,105) | (7,4,6,2) | ||
| (25,70,98,123) | (7,7,5) | (3,7,3) | (5,7,7,0) |
| (27,4,72,77) | (6,4,3,4) | ||
| (27,6,14,31) | (5,2,1,1) | ||
| (27,6,38,47) | (2,1,5) | (6,1,1,3) | |
| (27,6,74,79) | (2,7,5,1) | ||
| (27,8,24,37) | (4,4,2,1) | ||
| (27,12,44,53) | (2,2,5) | (6,2,2,3) | |
| (27,14,90,95) | (5,5,2) | (5,6,5,3) | |
| (27,16,96,101) | (8,0,1,6) | ||
| (27,18,34,47) | (6,1,3,1) | ||
| (27,24,68,77) | (2,4,5) | (6,4,4,3) | |
| (27,28,36,53) | (6,2,3,2) | ||
| (27,30,86,95) | (2,5,5) | (6,5,5,3) | |
| (27,34,66,79) | (7,2,5,1) | ||
| (27,40,60,77) | (5,6,4) | (4,6,5,0) | |
| (27,44,96,109) | (8,2,4,5) | ||
| (27,64,84,109) | (8,2,5,4) | ||
| (27,88,96,133) | (8,4,7,2) | ||
| (29,2,26,39) | (3,5,2,1) | ||
| (29,2,82,87) | (3,7,5,2) | ||
| (29,8,88,93) | (6,5,4,4) | ||
| (29,12,96,101) | (6,8,1) | (1,8,6,0) | |
| (29,14,22,39) | (5,3,2,1) | ||
| (29,20,28,45) | (6,1,2,2) | ||
| (29,24,48,61) | (4,6,3) | (3,6,4,0) | |
| (29,28,56,69) | (7,0,2,4) | ||
| (29,48,84,101) | (6,7,4) | (4,7,6,0) | |
| (31,2,94,99) | (7,4,3,5) | ||
| (31,8,8,33) | (1,4,4) | (4,4,1,0) | |
| (31,8,100,105) | (2,8,6,1) | ||
| (31,12,24,41) | (6,0,1,2) | ||
| (31,12,36,49) | (3,6,2) | (2,6,3,0) | |
| (31,14,38,51) | (4,5,3,1) | ||
| (31,22,34,51) | (5,4,3,1) | ||
| (31,38,86,99) | (8,1,3,5) | ||
| (31,40,92,105) | (8,2,6,1) | ||
| (31,42,42,67) | (7,0,3,3) | ||
| (31,46,82,99) | (7,4,5,3) | ||
| (31,58,74,99) | (8,1,5,3) | ||
| (31,90,90,131) | (9,0,5,5) | ||
| (33,4,24,41) | (2,6,1) | (1,6,2,0) | |
| (33,4,36,49) | (5,4,2,2) | ||
| (33,6,10,35) | (1,5,3) | (3,5,1,0) | |
| (33,6,58,67) | (2,1,6) | (7,1,1,4) | |
| (33,12,64,73) | (2,2,6) | (7,2,2,4) | |
| (33,18,74,83) | (2,3,6) | (7,3,3,4) | |
| (33,22,54,67) | (7,1,4,1) | ||
| (33,24,88,97) | (2,4,6) | (7,4,4,4) | |
| (33,30,50,67) | (5,5,4,1) | ||
| (33,30,70,83) | (5,7,3) | (3,7,5,0) | |
| (33,38,66,83) | (7,3,4,3) | ||
| (33,44,48,73) | (7,2,4,2) | ||
| (33,56,72,97) | (8,1,4,4) | ||
| (33,68,84,113) | (8,3,6,2) | ||
| (33,92,96,137) | (9,2,6,4) | ||
| (35,4,28,45) | (6,2,1,2) | ||
| (35,6,42,55) | (4,3,3) | (3,6,3,1) | |
| (35,10,14,39) | (3,1,4) | (6,1,1,1) | |
| (35,20,56,69) | (4,4,3) | (4,6,4,1) | |
| (35,30,54,71) | (7,2,3,3) | ||
| (35,32,80,93) | (8,0,2,5) | ||
| (35,38,70,87) | (4,5,3) | (5,6,5,1) | |
| (35,40,44,69) | (3,4,4) | (6,4,4,1) | |
| (35,50,62,87) | (3,5,4) | (6,5,5,1) | |
| (35,60,84,109) | (3,6,4) | (6,6,6,1) | |
| (35,72,96,125) | (8,4,6,3) | ||
| (35,86,98,135) | (4,7,3) | (7,6,7,1) | |
| (37,10,50,63) | (5,5,3,2) | ||
| (37,12,36,53) | (6,3,2,2) | ||
| (37,16,20,45) | (2,5,4) | (4,5,2,0) | |
| (37,16,56,69) | (4,7,2) | (2,7,4,0) | |
| (37,22,46,63) | (7,1,2,3) | ||
| (37,34,38,63) | (7,1,3,2) | ||
| (37,36,96,109) | (6,8,3) | (3,8,6,0) | |
| (37,46,94,111) | (5,7,6,1) | ||
| (39,2,18,43) | (5,4,1,1) | ||
| (39,4,12,41) | (1,6,2) | (2,6,1,0) | |
| (39,6,82,91) | (2,1,7) | (8,1,1,5) | |
| (39,12,88,97) | (2,2,7) | (8,2,2,5) | |
| (39,14,42,59) | (7,0,1,3) | ||
| (39,18,98,107) | (2,3,7) | (8,3,3,5) | |
| (39,20,48,65) | (6,4,3,2) | ||
| (39,48,64,89) | (8,0,3,4) | ||
| (39,52,72,97) | (8,2,5,2) | ||
| (39,54,62,91) | (7,4,5,1) | ||
| (39,62,78,107) | (8,3,5,3) | ||
| (41,6,42,59) | (3,7,1) | (1,7,3,0) | |
| (41,10,62,75) | (3,7,4,1) | ||
| (41,12,24,49) | (2,6,3) | (3,6,2,0) | |
| (41,16,68,81) | (5,6,4,2) | ||
| (41,28,28,57) | (7,0,2,2) | ||
| (41,28,64,81) | (6,5,4,2) | ||
| (41,30,30,59) | (3,5,5) | (5,5,3,0) | |
| (41,38,50,75) | (7,3,4,1) | ||
| (41,40,88,105) | (8,3,4,4) | ||
| (41,62,98,123) | (9,1,4,5) | ||
| (43,2,46,63) | (7,2,1,3) | ||
| (43,6,18,47) | (6,3,1,1) | ||
| (43,6,66,79) | (6,5,3,3) | ||
| (43,20,80,93) | (5,8,2) | (2,8,5,0) | |
| (43,24,36,61) | (3,6,4) | (4,6,3,0) | |
| (43,26,38,63) | (7,2,3,1) | ||
| (43,32,76,93) | (8,2,3,4) | ||
| (43,36,84,101) | (6,6,5,2) | ||
| (43,52,64,93) | (8,2,4,3) | ||
| (43,64,88,117) | (8,4,6,1) | ||
| (45,4,72,85) | (5,4,3) | (4,7,4,2) | |
| (45,10,54,71) | (7,3,2,3) | ||
| (45,18,26,55) | (7,1,2,1) | ||
| (45,20,36,61) | (3,2,5) | (7,2,2,2) | |
| (45,22,90,103) | (5,5,3) | (5,7,5,2) | |
| (45,24,68,85) | (8,1,2,4) | ||
| (45,30,46,71) | (3,3,5) | (7,3,3,2) | |
| (45,40,48,77) | (4,6,5) | (5,6,4,0) | |
| (45,50,78,103) | (3,5,5) | (7,5,5,2) | |
| (47,4,32,57) | (4,6,2,1) | ||
| (47,12,84,97) | (6,6,4,3) | ||
| (47,14,14,51) | (7,0,1,1) | ||
| (47,14,86,99) | (3,8,5,1) | ||
| (47,16,28,57) | (6,4,2,1) | ||
| (47,16,64,81) | (8,0,1,4) | ||
| (47,18,66,83) | (7,4,3,3) | ||
| (47,42,54,83) | (8,1,3,3) | ||
| (47,46,74,99) | (8,3,5,1) | ||
| (47,54,90,115) | (9,0,3,5) | ||
| (47,60,60,97) | (5,6,6) | (6,6,5,0) | |
| (49,2,14,51) | (1,7,1) | (4,1,4) | (1,7,1,0) |
| (49,2,86,99) | (7,5,3,4) | ||
| (49,8,28,57) | (2,7,2) | (4,2,4) | (2,7,2,0) |
| (49,8,64,81) | (4,8,1) | (1,8,4,0) | |
| (49,10,10,51) | (1,5,5) | (5,5,1,0) | |
| (49,18,42,67) | (3,7,3) | (4,3,4) | (3,7,3,0) |
| (49,26,82,99) | (7,5,4,3) | ||
| (49,32,56,81) | (4,7,4) | (4,4,4) | (4,7,4,0) |
| (49,50,70,99) | (5,7,5) | (4,5,4) | (5,7,5,0) |
| (49,72,72,113) | (9,0,4,4) | ||
| (49,72,84,121) | (6,7,6) | (4,6,4) | (6,7,6,0) |
| (49,76,92,129) | (8,5,6,2) | ||
| (51,8,12,53) | (1,6,4) | (4,6,1,0) | |
| (51,8,96,109) | (4,8,5,2) | ||
| (51,10,18,55) | (7,2,1,1) | ||
| (51,18,46,71) | (5,6,3,1) | ||
| (51,26,42,71) | (6,5,3,1) | ||
| (51,32,48,77) | (8,0,2,3) | ||
| (51,32,60,85) | (8,2,4,1) | ||
| (51,64,72,109) | (8,4,5,2) | ||
| (51,74,78,119) | (9,2,5,3) | ||
| (53,2,34,63) | (7,3,1,2) | ||
| (53,4,44,69) | (6,5,2,2) | ||
| (53,8,76,93) | (8,3,2,4) | ||
| (53,22,26,63) | (7,3,2,1) | ||
| (53,26,94,111) | (9,1,2,5) | ||
| (53,46,86,111) | (9,1,5,2) | ||
| (53,52,56,93) | (8,3,4,2) | ||
| (53,56,88,117) | (9,2,4,4) | ||
| (53,60,96,125) | (6,8,5) | (5,8,6,0) | |
| (55,16,88,105) | (8,4,3,4) | ||
| (55,18,90,107) | (9,0,1,5) | ||
| (55,20,56,81) | (3,2,6) | (8,2,2,3) | |
| (55,22,46,75) | (8,1,3,1) | ||
| (55,30,66,91) | (3,3,6) | (8,3,3,3) | |
| (55,34,38,75) | (7,4,3,1) | ||
| (55,36,60,89) | (6,6,4,1) | ||
| (55,40,44,81) | (8,2,3,2) | ||
| (55,50,98,123) | (3,5,6) | (8,5,5,3) | |
| (57,6,14,59) | (1,7,3) | (3,7,1,0) | |
| (57,10,90,107) | (5,9,1) | (1,9,5,0) | |
| (57,12,44,73) | (7,4,2,2) | ||
| (57,20,24,65) | (2,6,5) | (5,6,2,0) | |
| (57,24,64,89) | (4,8,3) | (3,8,4,0) | |
| (57,26,66,91) | (5,7,4,1) | ||
| (57,28,36,73) | (8,1,2,2) | ||
| (57,46,54,91) | (7,5,4,1) | ||
| (57,46,78,107) | (9,1,3,4) | ||
| (57,62,66,107) | (9,1,4,3) | ||
| (57,84,92,137) | (9,4,6,2) | ||
| (59,2,22,63) | (6,5,1,1) | ||
| (59,2,94,111) | (2,9,5,1) | ||
| (59,12,48,77) | (3,8,2) | (2,8,3,0) | |
| (59,12,60,85) | (6,6,3,2) | ||
| (59,16,32,69) | (8,0,1,2) | ||
| (59,38,86,111) | (9,2,5,1) | ||
| (59,46,82,111) | (6,7,5,1) | ||
| (59,58,74,111) | (7,6,5,1) | ||
| (59,80,100,141) | (10,0,4,5) | ||
| (61,2,62,87) | (5,7,3,2) | ||
| (61,4,32,69) | (2,8,1) | (1,8,2,0) | |
| (61,16,28,69) | (2,7,4) | (4,7,2,0) | |
| (61,22,58,87) | (7,5,3,2) | ||
| (61,36,72,101) | (9,0,2,4) | ||
| (61,66,78,119) | (9,3,5,2) | ||
| (61,70,98,135) | (7,7,6,1) | ||
| (63,2,54,83) | (5,3,4) | (3,8,3,1) | |
| (63,6,22,67) | (7,4,1,1) | ||
| (63,8,36,73) | (8,2,1,2) | ||
| (63,14,18,67) | (4,1,5) | (8,1,1,1) | |
| (63,16,72,97) | (5,4,4) | (4,8,4,1) | |
| (63,24,28,73) | (4,2,5) | (8,2,2,1) | |
| (63,34,42,83) | (4,3,5) | (8,3,3,1) | |
| (63,34,90,115) | (5,5,4) | (5,8,5,1) | |
| (63,48,56,97) | (4,4,5) | (8,4,4,1) | |
| (63,66,70,115) | (4,5,5) | (8,5,5,1) | |
| (63,66,94,131) | (9,4,5,3) | ||
| (63,84,88,137) | (4,6,5) | (8,6,6,1) | |
| (63,84,100,145) | (10,2,5,4) | ||
| (65,10,74,99) | (3,1,7) | (9,1,1,4) | |
| (65,20,44,81) | (8,3,2,2) | ||
| (65,26,70,99) | (9,1,4,1) | ||
| (65,30,42,83) | (3,7,5) | (5,7,3,0) | |
| (65,32,76,105) | (7,6,4,2) | ||
| (65,52,64,105) | (9,2,4,2) | ||
| (65,54,78,115) | (9,3,4,3) | ||
| (67,4,16,69) | (1,8,2) | (2,8,1,0) | |
| (67,6,78,103) | (7,6,3,3) | ||
| (67,8,64,93) | (8,4,2,3) | ||
| (67,12,36,77) | (6,6,2,1) | ||
| (67,32,56,93) | (8,4,3,2) | ||
| (67,42,66,103) | (9,2,3,3) | ||
| (67,70,94,135) | (10,1,5,3) | ||
| (69,6,38,79) | (5,7,2,1) | ||
| (69,8,84,109) | (5,8,4,2) | ||
| (69,12,32,77) | (2,8,3) | (3,8,2,0) | |
| (69,16,72,101) | (4,9,2) | (2,9,4,0) | |
| (69,18,34,79) | (7,5,2,1) | ||
| (69,30,58,95) | (9,1,2,3) | ||
| (69,42,50,95) | (9,1,3,2) | ||
| (69,42,98,127) | (7,7,5,2) | ||
| (69,44,72,109) | (8,5,4,2) | ||
| (69,48,56,101) | (4,7,6) | (6,7,4,0) | |
| (69,72,88,133) | (10,1,4,4) | ||
| (69,78,98,143) | (9,5,6,1) | ||
| (71,10,22,75) | (8,3,1,1) | ||
| (71,12,12,73) | (1,6,6) | (6,6,1,0) | |
| (71,18,54,91) | (9,0,1,3) | ||
| (71,18,78,107) | (8,5,3,3) | ||
| (71,22,98,123) | (4,9,5,1) | ||
| (71,24,48,89) | (3,8,4) | (4,8,3,0) | |
| (71,40,100,129) | (10,0,2,5) | ||
| (71,56,92,129) | (8,6,5,2) | ||
| (71,58,82,123) | (9,4,5,1) | ||
| (73,6,54,91) | (3,9,1) | (1,9,3,0) | |
| (73,6,78,107) | (3,9,4,1) | ||
| (73,10,14,75) | (1,7,5) | (5,7,1,0) | |
| (73,14,98,123) | (7,7,4,3) | ||
| (73,36,36,89) | (9,0,2,2) | ||
| (73,40,64,105) | (4,8,5) | (5,8,4,0) | |
| (73,42,66,107) | (9,3,4,1) | ||
| (73,70,70,123) | (5,7,7) | (7,7,5,0) | |
| (75,6,58,95) | (9,2,1,3) | ||
| (75,22,54,95) | (6,7,3,1) | ||
| (75,24,32,85) | (8,4,2,1) | ||
| (75,28,96,125) | (8,6,4,3) | ||
| (75,30,98,127) | (10,1,5,1) | ||
| (75,60,92,133) | (10,2,5,2) | ||
| (75,82,90,143) | (10,3,5,3) | ||
| (77,4,52,93) | (7,6,2,2) | ||
| (77,4,88,117) | (9,4,2,4) | ||
| (77,14,38,87) | (4,1,6) | (9,1,1,2) | |
| (77,18,66,103) | (9,3,2,3) | ||
| (77,22,34,87) | (9,1,2,1) | ||
| (77,28,44,93) | (4,2,6) | (9,2,2,2) | |
| (77,42,54,103) | (4,3,6) | (9,3,3,2) | |
| (77,56,68,117) | (4,4,6) | (9,4,4,2) | |
| (77,70,86,135) | (4,5,6) | (9,5,5,2) | |
| (77,84,96,149) | (6,8,7) | (7,8,6,0) | |
| (79,8,16,81) | (1,8,4) | (4,8,1,0) | |
| (79,14,58,99) | (5,8,3,1) | ||
| (79,18,18,83) | (9,0,1,1) | ||
| (79,20,100,129) | (5,10,2) | (2,10,5,0) | |
| (79,30,78,115) | (9,4,3,3) | ||
| (79,38,46,99) | (8,5,3,1) | ||
| (79,44,92,129) | (10,2,3,4) | ||
| (79,68,76,129) | (10,2,4,3) | ||
| (81,2,18,83) | (1,9,1) | (5,1,5) | (1,9,1,0) |
| (81,8,36,89) | (2,9,2) | (5,2,5) | (2,9,2,0) |
| (81,12,52,97) | (8,5,2,2) | ||
| (81,24,28,89) | (2,7,6) | (6,7,2,0) | |
| (81,32,72,113) | (4,9,4) | (5,4,5) | (4,9,4,0) |
| (81,32,84,121) | (10,1,2,4) | ||
| (81,42,70,115) | (7,7,4,1) | ||
| (81,42,94,131) | (9,5,4,3) | ||
| (81,48,76,121) | (10,1,4,2) | ||
| (81,50,90,131) | (5,9,5) | (5,5,5) | (5,9,5,0) |
| (83,2,26,87) | (7,6,1,1) | ||
| (83,14,22,87) | (9,2,1,1) | ||
| (83,20,80,117) | (10,0,1,4) | ||
| (83,32,76,117) | (6,8,4,1) | ||
| (83,50,94,135) | (10,3,5,1) | ||
| (83,52,64,117) | (8,6,4,1) | ||
| (83,54,66,119) | (10,1,3,3) | ||
| (85,6,42,95) | (9,3,1,2) | ||
| (85,8,80,117) | (4,10,1) | (1,10,4,0) | |
| (85,14,70,111) | (7,7,3,2) | ||
| (85,20,32,93) | (2,8,5) | (5,8,2,0) | |
| (87,4,72,113) | (6,8,3,2) | ||
| (87,4,84,121) | (10,2,1,4) | ||
| (87,6,26,91) | (8,5,1,1) | ||
| (87,6,62,107) | (4,9,3,1) | ||
| (87,24,68,113) | (8,6,3,2) | ||
| (87,36,76,121) | (10,2,4,1) | ||
| (87,40,60,113) | (10,0,2,3) | ||
| (87,42,46,107) | (9,4,3,1) | ||
| (87,54,94,139) | (7,8,5,1) | ||
| (87,66,86,139) | (8,7,5,1) | ||
| (87,80,84,145) | (10,4,5,2) | ||
| (89,6,18,91) | (1,9,3) | (3,9,1,0) | |
| (89,16,92,129) | (10,3,2,4) | ||
| (89,20,52,105) | (9,4,2,2) | ||
| (89,22,82,123) | (5,9,4,1) | ||
| (89,42,42,107) | (3,7,7) | (7,7,3,0) | |
| (89,58,62,123) | (9,5,4,1) | ||
| (89,64,68,129) | (10,3,4,2) | ||
| (89,88,88,153) | (11,0,4,4) | ||
| (91,14,62,111) | (4,1,7) | (10,1,1,3) | |
| (91,26,58,111) | (10,1,3,1) | ||
| (91,28,68,117) | (4,2,7) | (10,2,2,3) | |
| (91,36,48,109) | (3,8,6) | (6,8,3,0) | |
| (91,40,100,141) | (5,10,4) | (4,10,5,0) | |
| (91,42,78,127) | (4,3,7) | (10,3,3,3) | |
| (91,56,92,141) | (4,4,7) | (10,4,4,3) | |
| (91,86,98,159) | (11,2,5,3) | ||
| (93,6,74,119) | (9,5,2,3) | ||
| (93,14,42,103) | (7,7,2,1) | ||
| (93,16,36,101) | (2,9,4) | (4,9,2,0) | |
| (93,24,80,125) | (4,10,3) | (3,10,4,0) | |
| (93,24,92,133) | (7,8,4,2) | ||
| (93,34,66,119) | (9,5,3,2) | ||
| (93,36,44,109) | (10,1,2,2) | ||
| (93,36,88,133) | (8,7,4,2) | ||
| (95,6,90,131) | (8,7,3,3) | ||
| (95,8,44,105) | (6,8,2,1) | ||
| (95,10,26,99) | (9,4,1,1) | ||
| (95,12,60,113) | (3,10,2) | (2,10,3,0) | |
| (97,4,40,105) | (2,10,1) | (1,10,2,0) | |
| (97,12,96,137) | (6,9,4,2) | ||
| (97,14,14,99) | (1,7,7) | (7,7,1,0) | |
| (97,30,54,115) | (3,9,5) | (5,9,3,0) | |
| (97,48,84,137) | (9,6,4,2) | ||
| (97,56,64,129) | (4,8,7) | (7,8,4,0) | |
| (97,58,94,147) | (11,1,3,4) | ||
| (97,74,82,147) | (11,1,4,3) | ||
| (99,2,66,119) | (10,3,1,3) | ||
| (99,12,16,101) | (1,8,6) | (6,8,1,0) | |
| (99,12,44,109) | (10,2,1,2) | ||
| (99,12,88,133) | (6,4,5) | (4,10,4,1) | |
| (99,18,22,103) | (5,1,6) | (10,1,1,1) | |
| (99,28,36,109) | (5,2,6) | (10,2,2,1) | |
| (99,38,54,119) | (5,3,6) | (10,3,3,1) | |
| (99,52,72,133) | (5,4,6) | (10,4,4,1) | |
| (99,70,90,151) | (5,5,6) | (10,5,5,1) | |
| 530個 | 104/530 19.6226% | 137/530 25.8491% | 530/530 100% |
パターン3は優秀でした。
パターン3でも、nやqに0を許さず、m,n,p,q∈Nとしてしまうと
383/530で72.2642%となってしまいます。
ではでは