P^2+Σ2P

The smallest pair of amicable numbers is (220, 284). You expand it infinitely. M and N are amicable. σ(M) is the sum of proper divisors. σ(220)=1+2+4+5+10+11+20+22+44+55+110+220=504. Moreover, σ(N)=σ(284)=1+2+4+71+142+284=504. σ(M)-M=N and σ(N)-N=M In this case, M=220 and N=284. σ(220)-220=504-220=284 σ(284)-284=504-284=220 ∴ σ(M)=M+N=σ(N) This is crossing over. M=apq, N=ar pqr are different prime numbers, and a is the prime relatively. Leonhard Euler found it. σ(p)=p+1, p is the prime. σ(pq)=σ(p)σ(q)=1+p+q+pq=(1+p)+q(1+p)=(1+p)(1+q), p and q are prime numbers. Therefore, σ(M)=σ(N), σ(apq)=σ(ar) σ(a)σ(p)σ(q)=σ(a)σ(r) σ(p)σ(q)=σ(r) ∴ (p+1)(q+1)=r+1 This is based on prime numbers. Then, you put x=p+1, y=q+1 . xy=r+1, r=xy-1 σ(M)=M+N=apq+ar=a(pq+r) σ(M)=σ(a)σ(p)σ(q)=σ(a)(p+1)(q+1)=a(pq+r) σ(a)xy=a[(x-1)(y-1)+(xy-1)] ax=[2ax-σ(a)x-a]y y=ax/[2ax-σ(a)x-a] Then, you put a/[2a-σ(a)]=b/c . 2a-σ(a)=ac/b, σ(a)=2a-(ac/b) ∴ y=ax/[(ac/b)x-a]=bx/(cx-...

Euler's formula is based on Maclaurin's expansion . The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin. x is 0, so f(x)=(sin x,cos x,-sin x,-cos x,・・・)=(0,1,0,-1,・・・)=sin x and f(x)=(cos x,-sin x,-cos x,sin x,・・・)=(1,0,-1,0,・・・)=cos x Euler's identity is well known as the beauty of mathematics.