5 and 12 Nov 2020
Jerzy Kocik
Integrality implies super-integrality and hyper-integrality
Part 2: Disks as vectors in Minkowski space, inversions, and conclusion of the proof.
See Apolloniana for some background
Recording of Part 1
Recording of Part 2
Part 1: Here are two warm-up problems (related, but not central):
Suppose 4 integers do not have a common factor, gcd(a,b,c,d)=1, and satisfy quadratic equation $$(a+b+c+d)^2 = 2\, (a^2+b^2+c^2+d^2)$$ Show that:
1. Exactly two of them are odd (easy)
2. The sum of any two can be written as a sum of two squares (a challenge)
Examples of solutions to the quadratic equation: (2,3,6,23), (2,3,15,38), (3,6, 14 47), (0,1,4,9), (11,14,15,89),...
Part 1: Here are two warm-up problems (related, but not central):
Suppose 4 integers do not have a common factor, gcd(a,b,c,d)=1, and satisfy quadratic equation $$(a+b+c+d)^2 = 2\, (a^2+b^2+c^2+d^2)$$ Show that:
1. Exactly two of them are odd (easy)
2. The sum of any two can be written as a sum of two squares (a challenge)
Examples of solutions to the quadratic equation: (2,3,6,23), (2,3,15,38), (3,6, 14 47), (0,1,4,9), (11,14,15,89),...
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