Expert: Sherry Wallin - 5/30/2011

Question
((3*n + 1 = k**2)   and
(4*n + 1 = m**2))  implies n is divisible by 56

I can show that n is divisible by 8, but I fail to show divisibility by 7. Can you please help?

Answer
Helmut~

I see a couple of things I will need clarified. I am not certain what your asterisks (*) mean. At first I thought multiplication but why use a single asterisk on one side and a double asterisk on the other?? If in fact they do just mean multiplication then what you really have is:

3n+1 = 2k and 4n+1 = 2m implies n is divisible by 56.

This is impossible since 3n+1 = 2k says that 3n+1 is even since any even number can be written as 2 times something. The only way 3n + 1 will be even is if n is odd. (Try an even number for n and you will see that you have 3 times an even number which will be even and when you add a 1 to the result you will then have an odd number contradicting that 3n+1 is even). This means that n has to be odd, hence n cannot be divisible by 56. (no even number will ever divide an odd number).

And 4n+1 = 2m says that 4n+1 is an even number. Sadly to say 4n+1 is an odd number, so it cannot be 2m. (If 4n+1 was even it has to be able to be written as 2 times something. Let 4n+1 = 2a where a is an integer. Now then, 2a = 4n + 1 simplifies to 2a - 4n = 1 which in turn implies that 2(a-2n) = 1. On the left is an even number (2 time something) equal again to an odd number which is 1 in this case.

So you see if what you mean with your asterisks is 3n+1 = 2k and 4n+1 = 2m implies n is divisible by 56, this can't be proven because it just isn't true. Always look as best you can when doing a proof if is seems plausible because you can NOT prove something that isn't true.

If you meant something else by your * and ** please let me know so I can still help you.

Math Prof


PS I don't think you were able to show n is divisible by 8 because n is an odd number (as I have shown above)!

***Incidentally you put as your subject that you were finding n but your problem statement is about showing divisibility of n