Solve Index Arithmetic Problem: ax^4≡2 (mod 13)

[buzzmath]

How would i go about solving the problem of for which values of a is the congruence ax^4≡2(mod 13) solvable? I think it might have something to do with power residues but I'm not sure.
Thanks

 

[ramsey2879]

How would i go about solving the problem of for which values of a is the congruence ax^4≡2(mod 13) solvable? I think it might have something to do with power residues but I'm not sure.
Thanks

I would first find all values of x^2 mod 13 for x = 1 to 6 since the possible values just repeat for x > 6. For x = 1,2 and 4 they are 1,4 and 3, respectively. For x = 3,5 and 6 they are -4, -1 and -3. Squaring those values give just three possible values of x^4, i.e., 1,9,3 having respective $$a$$ values of 2, 6 and 5.
If there is an easilier way, let someone else post it. Note that 5*3 = 2 mod 13, 2*3 = 6 mod 13 and 6*3 = 5 mod 13 so I guess that powers of 3 are significant here.

 

[aujing]

oops,, what is that "ax^4≡2(mod 13)" means ??
ax^4=13n+2 !?
how do you use this "mod" stuff,, I don't use it in this format,, because excel and VBA not in this format... ...

 

[Gokul43201]

$$a \equiv b ~(mod~n)$$ means n divides a-b.

Congruences modulo the same number (n above) can be added, subtracted or multiplied together, just like regular equations.

Can't say I understand your "format" question.

Many programming languages (and possibly Excel) include a function along the lines of "mod(a,n)" which usually returns the smallest positive b, such that b == a (mod n).