MHB Finding Primitive Roots Modulo 169: Is There a Smarter Way?

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To find a primitive root modulo 169, it is established that either 2 or 15 can serve as a primitive root. Testing 2 is the first step; if it does not work, then 15 is the alternative. The discussion emphasizes that checking all orders dividing φ(13²) is unnecessary; instead, one should verify orders derived from φ(13²) divided by distinct primes. Since 2 is a primitive root mod 13, its order mod 169 will either be 12 or 156, simplifying the verification process. Ultimately, confirming whether 2 is a primitive root mod 169 eliminates the need to check other powers.
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How can I find a primitive root modulo 169?
I found the primitive roots mod 13 by testing 2, and then concluding that any 2^k with (k, 12)=1 would do. So that gave me 2, 6, 7 and 11. But modulo 13 I have no idea how to start.. I’m sure there’s a smarter way than trying 2^the orders that divide phi(13^2)..?.
 
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tda120 said:
How can I find a primitive root modulo 169?
I found the primitive roots mod 13 by testing 2, and then concluding that any 2^k with (k, 12)=1 would do. So that gave me 2, 6, 7 and 11. But modulo 13 I have no idea how to start.. I’m sure there’s a smarter way than trying 2^the orders that divide phi(13^2)..?.
Hi tda, and welcome to MHB. You might be interested in this link, which tells you that the answer to your question is either $2$ or $2+13=15$. That still leaves you with the work of testing to see if $2$ works. If it does not, then $15$ does.
 
tda120 said:
How can I find a primitive root modulo 169?

There is no simple method. You can cobble up together some basic theories on primitive roots, find a bit of a rough upperbound (although none is known to be useful for small cases) and some modular exponentiation to get a fast enough algorithm.

As Opalg there indicated, that either 2 or 15 is primitive root modulo 169, can easily be found for this case. Try proving the former and then the later if it doesn't work.
 
tda120 said:
How can I find a primitive root modulo 169?
I found the primitive roots mod 13 by testing 2, and then concluding that any 2^k with (k, 12)=1 would do. So that gave me 2, 6, 7 and 11. But modulo 13 I have no idea how to start.. I’m sure there’s a smarter way than trying 2^the orders that divide phi(13^2)..?.

You don't have to check all the orders that divide $\phi(13^2)$.
It suffices to check each of the orders that are $\phi(13^2)$ divided by one of the distinct primes it contains.

In your case:
$$\phi(13^2)=2^2\cdot 3 \cdot 13$$
So the orders to verify are:
$$2\cdot 3 \cdot 13,\quad 2^2 \cdot 13, \quad 2^2\cdot 3$$
 
Opalg said:
Hi tda, and welcome to MHB. You might be interested in this link, which tells you that the answer to your question is either $2$ or $2+13=15$. That still leaves you with the work of testing to see if $2$ works. If it does not, then $15$ does.

Nice!

From that link we also get that since 2 is a primitive root mod 13, it follows that the order of 2 mod 169 is either (13-1) or 13(13-1).
So if $2^{13-1} \not\equiv 1 \pmod{169}$ that means that 2 has to be a primitive root mod 169. Or otherwise 15 has to be.

In other words, no need to check any of the other powers.
 
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