I took 5 - but that gave 153 - where is the hint!
Modulo 5.I took 5 - but that gave 153 - where is the hint!
3=Tnmod5 for all n≥4?Modulo 5.
I have to admit that I tried it by considering lots and lots of cases before I found your proof with seven and then thought: if seven works, then five probably, too. The final proof was indeed a little beauty of only three lines.Hint: Perfect squares do not end in a 3. When n=4 you get 33.
I'll give you another hint tomorrow as the next hint will give everything away.
The proof is sweet--and I did it on my own!
I guess that I'm wrong but I would think that (3/5) = 3*5-1(mod 5). Doesn't 5-1(mod 5) = 0(mod5). If so, then (3/5) = 0(mod 5) and not 9. Where is my error?Here is mine:
Sn:=j=1∑nj!≡3(mod5) and (53)≡32≡−1(mod5)Hence Sn is no square for n≥4, and since S2=3, there are no other squares than S1=12,S3=32.
(pa) is the Legendre symbol in this case and not a quotient of integers.I guess that I'm wrong but I would think that (3/5) = 3*5-1(mod 5). Doesn't 5-1(mod 5) = 0(mod5). If so, then (3/5) = 0(mod 5) and not 9. Where is my error?