Number theory - equation.

[platinum983]

Find all (n,k) whole numbers for
n⁴-1=80*34^k
I have no idea where to start in solving this.

 

[Dr.Peterson]

I would start by factoring the LHS and the RHS.

 

[JeffM]

There is a good reason for the name Fundamental Theorem of Arithmetic.

Factor the LHS as Dr. Peterson advised, and factor the RHS into powers of primes.

 

[Deleted member 4993]

Find all (n,k) whole numbers for
n⁴-1=80*34^k
I have no idea where to start in solving this.

Please clarify whether the problem is:

n⁴-1 = 80 * (34^k)

Or

n⁴-1 = (80 * 34)^k

in either case, observe that the RHS is an even number and the 'n' in the LHS is an odd number.

 

[pka]

I frankly think there is a misprint in this question.
I could see it being: Find all (n,k) whole numbers for n⁴=80*34^k
so that \(80\cdot 34^k=2^{5+k}\cdot 5\cdot 17^k\)
BUT \(n^4-1=(n-1)(n+1)(n^2+1)\) which has little to do with exponents.
Please check the OP for a misprint and advise.

 

[platinum983]

There was no misprint, 80 * (34^k) on right side, and somehow I need to prove that k must be 0.

 

[Steven G]

Based on pka's factoring in post #5 what are the possible values for k? Just show that all possible values for k other than 0 will not work.
In order to receive further help you must show some work.

 

[pka]

There was no misprint, 80 * (34^k) on right side, and somehow I need to prove that k must be 0.

Well then \(n=3~\&~k=0\) will work. Proof???