# I need help with this divisibility problem.

I need help with the following divisibility problem. Find all prime numbers m and n such that $mn |12^{n+m}-1$ and $m= n+2$.

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What theorems do you know that could be helpful for this problem? – Phira Apr 9 '12 at 8:47
$(5,7),(11,13),(29,31)$ – pedja Apr 9 '12 at 8:48
You have primes $p$ and $p+2$, and you want to choose $p$ so that $p(p+2)\mid 12^{2p+2}-1=\left(12^{p+1}-1\right)\left(12^{p+1}+1\right)$. – Brian M. Scott Apr 9 '12 at 8:55

You want to solve $p(p+2)|(12^{p+1}-1)(12^{p+1}+1)$.

Hint: First exclude $p=2,3$, so we have \eqalign{ 12^{p+1}-1 \equiv 143 &= 11 \cdot 13 &\pmod p,\\ 12^{p+1}+1 \equiv 145 &= 5 \cdot 29 &\pmod p, } and deduce that $p$ must be one of $5,11,29$.

Edit: I'll just add more details: We want that $p$ divides $(12^{p+1}-1)(12^{p+1}+1)$, so $p$ must divide one of the factors of this product. Suppose $p|12^{p+1}-1=k\cdot p+143$ (the congruence follows from Fermat's little theorem). This means $p|143$ and hence $p=11$ or $p=13$. If $p+2$ is prime then we automatically have $p+2|12^{p+1}-1$ again by Fermat's theorem, so $p=11$ is a solution. $p=13$ isn't, as $p+2$ is not prime.

In the other case, $p|12^{p+1}+1$ we get 2 solutions, $p=5$ and $p=29$.

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Something's wrong with the tex-compiler. I wanted to insert a linebreak into the equations, which is properly displayed in the preview, but not here. Does anybody find my mistake? – Michalis Apr 9 '12 at 9:41
I think I fixed the TeX, but I'm not sure which moduli your congruences are using. You can use \pmod{...} for that. – bgins Apr 9 '12 at 9:42
I don't see how you got $143$, could you elaborate? – Gigili Apr 9 '12 at 9:43
@Gigili: $12^2-1|12^{2k}-1$ for all $k\in\mathbb{N}$ – bgins Apr 9 '12 at 9:44
Ah yes, thank you @bgins. – Gigili Apr 9 '12 at 9:46

Here are some thoughts to start off...

The first twin primes are $(n,m)\in\{(3,5),(5,7),(11,13),(17,19)\}$. Clearly $3\not\mid12^{m+n}-1$ since $12^{m+n}\equiv0\pmod3$. Furthermore, $12^{m+n}\equiv1\pmod{m,n}\implies2,3\not\mid m,n$ and $m+n\equiv0\pmod{m-1,n-1}$, which seems like it could disqualifies many candidates.

However, $11\cdot13=12^2-1\mid12^{24}-1$ so $(11,13)$ is a solution...

If $m=6k+1,~n=6k-1$ is prime, then $m-1=6k\mid m+n=12k$ $\implies$ $12^{12k}\equiv1\pmod m$ for free (from Euler's theorem), while $12^{12k}\equiv1\pmod n$ iff $\operatorname{ord}_n12\mid12k$.

Now

• for $k=1$, $\operatorname{ord}_512=4\mid12$;
• for $k=2$, $\operatorname{ord}_{11}12=1\mid24$;
• for $k=3$, $\operatorname{ord}_{17}12=16\not\mid36$;
• for $k=4$, $m$ is not prime;
• for $k=5$, $\operatorname{ord}_{29}12=4\mid12\cdot5$;
• for $k=7$ (the next twin prime pair), $\operatorname{ord}_{41}12=40\not\mid12\cdot7$.

Perhaps there is a good argument why all solutions must be of the form $6k\pm1$. A little checking in sage shows that @Pedja's solutions, $k=1,2,5$, are the only ones of this form for $k\le10^6$.

for k in range(1,100):
m = 6*k+1
n = 6*k-1
if is_prime(m) and is_prime(n):
test_order = multiplicative_order(mod(12,6*k-1))
test_value = 12*k % test_order
print '%5d%5d%5d%5d%5d' % (k, m, n, test_order, test_value)

1 4
2 1
5 4


In fact, there is a good argument using Fermat's little theorem, and @Michalis presents it in his post.

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