/sci/ - Science & Math

[Anonymous No.14531135]

Let a(1)=b(1)=c(1)=1
For all i>1 let a(i)=2*b(i-1)+a(i-1) and
b(i-1)=((a(i-1)-2b(i-1))^2-1)/2
Now, it's easy to see a(i),b(i),b(i+1) form a promitive Pythagorean triple. All kinds of cool things arise from this but i haven't found any reference. E.g. a(i) appears to be prime. That's my main concern and i haven't been able to prove that yet but it's easy to check with few lines of code. The geomrtry from which i derived this is the following. Take the 3,4,5 PT. Now extend a square from the short leg. On the opposite side of this square extend the same triangle. Now extend a square with side 2*4+3 to the bottom of the produced triangle and continue like that. Fun fact: if you inscribe circles in the first 2 squares, the ratio of the radius of the second to the first is equal to the ratio of the volumetric mean radius of the earth to that of the moon.

[Anonymous No.14531186]

Correction:
Now, it's easy to see a(i),b(i),b(i)+1 form a promitive Pythagorean triple.