Can this interesting property be proven?
$$2^2+3^2+5^2+7^2+9^2+11^2=(17)^2$$ $$22^2+33^2+55^2+77^2+99^2+11^2=(143)^2$$
Also:
$$22^2+33^2+55^2+77^2+99^2+121^2=(187)^2$$ $$222^2+333^2+555^2+777^2+999^2+1221^2=(1887)^2$$ $$2222^2+3333^2+5555^2+7777^2+9999^2+12221^2=(18887)^2$$ $$22222^2+33333^2+55555^2+77777^2+99999^2+122221^2=(188887)^2$$
How can I prove this?
Are there other interesting properties like these?
Solution 1:
Hint: What happens if you multiply your first identity with $111\cdots1^2$?