- #1
futb0l
I am having trouble with this one...
Prove that any positive integer whose ALL digits are 1s (except 1) is not a perfect square.
Prove that any positive integer whose ALL digits are 1s (except 1) is not a perfect square.
futb0l said:i seem to have another proof, every odd perfect square when divided by 8 has a remainder of 1... and noting that if an odd integer is squared the answer is always odd.
and with the exception of 1,11, every positive integer whose all digits are 1 have a remainder of 7. But 11 is clearly not a perfect square.
I think this should be enough proof..
but please explain Matt Grime's proof, because I'd like to learn.
futb0l said:I am having trouble with this one...
Prove that any positive integer whose ALL digits are 1s (except 1) is not a perfect square.
So if you add an even number with an odd number the answer is always an
odd number. So Tn is always an odd number. So an odd number can never be a square.
...this is the way to prove that all the positive integers where digits are 1s (except 1) is not a perfect square.
What does tihis have to do with whether 111... 111 is a perfect square?grabateetrap said:does an infinite string have an end?
Is this proven by noting the square of any any single digit no. would yield a 1 in the units' place only if it were 9 or 1? Is there a proof for this, or is this self-evident?Gokul43201 said:Now for the units' digit of x^2 to be 1, the units digit of x must be 1 or 9.
Dick said:If you work out p(mod 12) for p a prime, what are the possibilities? Excepting 2 and 3. What does this imply for p^2(mod 12)?