【数セミ】エレガントな解答をもとむ 4【2021.08】

**１３２人目の素数さん** · 2021/07/13(火) 00:00:20.12

締切りの過ぎた問題をみんなで議論しましょう。
ただし難易度情報は可。

過去スレ：
1. http://rio2016.5ch.net/test/read.cgi/math/1295154182/
2. http://rio2016.5ch.net/test/read.cgi/math/1476702312/
3. http://rio2016.5ch.net/test/read.cgi/math/1537116043/

**１３２人目の素数さん** · 2024/05/01(水) 01:00:00.65

a<b<c<d.
{a,b,c,d}.

4abcd+2ab+2ac+2ad+2bc+2bd+2cd-a^2-b^2-c^2-d^2-4=0.

h(p,q,r,s)=4pqr+2p+2q+2r-s.

a<b<c<d.
a<h(b,c,d,a).
b<h(a,c,d,b).
c<h(a,b,d,c).
h(a,b,c,d)<d.

A({a,b,c,d})={h(b,c,d,a),b,c,d}.
B({a,b,c,d})={a,h(a,c,d,b),c,d}.
C({a,b,c,d})={a,b,h(a,b,d,c),d}.
D({a,b,c,d})={a,b,c,h(a,b,c,d)}.

xy+1=z^2.
S(x,y)={0,x,y,x+y+2z}.

D({1,3,8,120})={0,1,3,8}=S(1,3).

D(D(D(D(D({81510,1643405480,225042690820638808,120581457028771819044676558804776})))))
=D(D(D(D({81510,1643405480,225042690820638808,420}))))
=D(D(D({81510,1643405480,12,420})))
=D(D({81510,4,12,420}))
=D({2,4,12,420})
={2,4,12,0}
=S(2,4).

mn(mn-2)+1=(mn-1)^2.
A(S(m,n(mn-2)))={m,n(mn-2),(n+1)(mn+m-2),4(mn-1)(mn+m-1)(mn^2+mn-2n-1)}.

mn(mn+2)+1=(mn+1)^2.
A(S(m,n(mn+2)))={m,n(mn+2),(n+1)(mn+m+2),4(mn+1)(mn+m+1)(mn^2+mn+2n+1)}.

xy+1=z^2.
A(S(x,y))={x,y,x+y+2z,4z(x+z)(y+z)}.

P^2=ah(a,b,c,d)+1.
Q^2=4a^2(bc+1).
R^2=ad+1.
(P+Q+R)(Q+R-P)(P+R-Q)(P+Q-R)=4a^2(4abcd+2ab+2ac+2ad+2bc+2bd+2cd-a^2-b^2-c^2-d^2+4).