SOLUTION: Conjecture: Find all positive integers 'a' such that there exists an integer 'm' with the property that a|(m^2+1) and a|((m+1)^2 +1). Hint: First show that 'a' must also divide

Question 507684: Conjecture: Find all positive integers 'a' such that there exists an integer 'm' with the property that a|(m^2+1) and a|((m+1)^2 +1).
Hint: First show that 'a' must also divide 2m+1.
I know that when m divides n it can be defined as n=(m)(q). So in this case it would be (m^2+1)=(a)(q), and so forth with the other problems. I'm not sure if this is correct but since (m^2+1)=(a)(q) and ((m+1)^2 +1)=(a)(q) so I set
(m^2+1)=((m+1)^2 +1). I then foiled, moved everything to one side and simplified and got 2m+1 with is relevant to the hint, I think. I don't know if I'm going in the right direction but I've seem to have hit a wall. Also, I don't really know how to show that a|2m+1. Please help. Thanks.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
You somehow set m^2 + 1 and (m+1)^2 + 1 equal to each other, which cannot be true.

We can definitely say that a|(2m+1), but this is a weaker statement. Fortunately we can conclude that a is odd. In addition, we can state that a|(m^2 - 2m), by using a modular arithmetic argument:

So this means that a divides 2m+1 and a divides m(m-2). This is actually quite useful, because not many values of a satisfy. We can say that

Since 2m+1 and 12m+1 are congruent mod a, it follows that their difference (10m) is congruent to 0 mod a. This means either a divides m, or a = 5 (remember a cannot equal 2). a = 5 definitely works (try m = 2, 12, ...). If a divides m, then the only value of a is 1 (since a divides 2m+1). Thus a = 1 and a = 5 are the only solutions.

RELATED QUESTIONS
For arbitrary positive integers k, m, and n, will there exist a prime p such that... (answered by Edwin McCravy)

There exists an integer 'a' such that if a|2m+1 and/or a|(m^2+1) and/or (answered by richard1234)

The number of all pairs (m,n) are positive integers such that 1/m + 1/n + 1/mn = 2/5... (answered by MathLover1,ikleyn)

Find all pairs (m, n) of positive integers such that m2/(2mn2 - n3 + 1) is a positive... (answered by Edwin McCravy)

Can anyone help me with this integer problem? I have this problem but am unsure what it... (answered by CubeyThePenguin)

If f is a function such that for all integers m and n, f(m,1) = m+1 and f(m,n) =... (answered by MathLover1)

A function f(n) defined for all positive integers has the property that f(m) + f(n) =... (answered by Edwin McCravy)

Perfect Squares. Find a positive integer such that the integer increased by 1 is a... (answered by 303795)

The function f(n) takes the integers to the real numbers such that f(m + n) + f(m - n) = (answered by CPhill,ikleyn)