Prove that an integer with digits '1' is not a perfect square.

[Dick]

Hi Dick,

The problem stated that p is a prime number that is greater than 3. When you work out p^2(mod 12) you get a remainder of 1 for every prime number square that is greater than 3. This is not the case with 2 or 3. I was wondering if there was some proof behind this.

A direct way to see this is that p(mod 12) can only be 1,5,7 or 11. Otherwise it is divisible by 2 or 3. But 1^2, 5^2, 7^2 and 11^2 are all equal to 1 mod 12. This is what matt was alluding to when he said all primes are equal to +1 or -1 mod 6.

 

[mikeyflex]

For mikeyflex: every prime not equal to 2 or 3 is of the form 6m+1 or 6m-1 for some integer m. This completely explains your observation.

Thanks Matt,

The problem stated that when p is a prime greater than 3, to evaluate p^2 mod 12.

When you plug in prime values for p^2 mod 12 you get 1, which is an equivalence relation. Is there any significance to this problem as pertaining to a proof of sorts?

Thanks again Matt

 

[matt grime]

Did you actually read what I wrote and think about it for any length of time. Your observation is completely proven by what I wrote.

 

[matticus]

to explain what matt's saying much farther than what needs to be done:
every prime greater than 3 can be written as 6k +1 or 6k -1 (which can be shown easily by a proof by cases).

(6k+1)^2 = 36k^2 + 12k + 1 = 12(3k^2 + k) + 1 = 1mod12
(6k - 1)^2 = 36k^2 -12k +1 = 12(3k^2 - k) + 1 = 1mod12

that's really all the "signifigance" involved in the observation

 

[mikeyflex]

to explain what matt's saying much farther than what needs to be done:
every prime greater than 3 can be written as 6k +1 or 6k -1 (which can be shown easily by a proof by cases).

(6k+1)^2 = 36k^2 + 12k + 1 = 12(3k^2 + k) + 1 = 1mod12
(6k - 1)^2 = 36k^2 -12k +1 = 12(3k^2 - k) + 1 = 1mod12

that's really all the "signifigance" involved in the observation

Ahh, now I get it, and that must be the reason why this proof doesn't work for prime numbers 2 and 3. Thank you for the observation. Number theory never seems light the bulb for me.