Number theory, modulus of a large number

[jman2807]

I am having a bit of trouble with the following problem:

Find 3[sup:y9g2m5pw](F5-1)[/sup:y9g2m5pw] mod F5 where F5 is the fifth fermat number (2[sup:y9g2m5pw]2^5[/sup:y9g2m5pw] + 1).

I tried using euler's theorem that if gcd(a,n) = 1 then a[sup:y9g2m5pw]phi(n)[/sup:y9g2m5pw] = 1 (mod n) which allows me to reduce it down to

3[sup:y9g2m5pw]6701057[/sup:y9g2m5pw] mod F5 but I cannot seem to reduce it down any further and this number is still too large to work with. We are allowed to use a computer to do this problem. Any help is greatly appreciated,

 

[galactus]

Using your \(\displaystyle 3^{6701057}\equiv (mod \;\ 4294967297)\)

and using Maple, I got 497143886

 

[jman2807]

Re: Number theory, modulus of a large number (solved)

Thanks for the help. I actually miscalculated it a bit. I was one off so it should have been 3[sup:3ipbned4]6701056[/sup:3ipbned4] then with your suggestion of maple it comes out to 3029026160. Which i also verified using a mod p calculator http://www.mtholyoke.edu/~mpeterso/Appl ... pplet.html