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How to prove that two exponential terms are congruent to 7?

  1. Mar 9, 2017 #1
    1. The problem statement, all variables and given/known data
    "Prove: ##∀n∈ℕ,7|[3^{4n+1}-5^{2n-1}]##"

    2. Relevant equations


    3. The attempt at a solution
    (1) "We take the trivial case: ##n=1##, and notice that ##3^5-5=238## and ##7|238## because ##7⋅(34)=238##."

    (2) "Now let ##n=k## for some ##1<k∈ℕ##. Then we assume that ##7|[3^{4k+1}-5^{2k-1}]##. Now we must prove that ##7|[3^{4(k+1)+1}-5^{2(k+1)-1}]##. We rewrite the quantity as ##3^5(3^{4k})-5(5^{2k})##."

    Here is where I got stuck. Basically, I was going to express everything in terms of ##mod5## and say that ##3^5(3^{4k})-5(5^{2k})=2mod5##.

    "Now we have ##3^5(3^{4k})-5(5^{2k})=(3mod5)(1mod5)-(0mod5)(0mod5)=3mod5≠2mod5##."

    I don't know how to approach the problem starting from the first sentence of (2). Can anyone help? Thanks.
     
  2. jcsd
  3. Mar 9, 2017 #2

    fresh_42

    Staff: Mentor

    Hint: use ##(a^4x-b^2y)=(a^2x-by)(a^2+b)-a^2b(x-y)##.
     
  4. Mar 9, 2017 #3
    Okay, I got ##(3^5⋅9^k-5⋅5^k)(9^k+5^k##. I'm not sure how I would proceed by induction.
     
  5. Mar 9, 2017 #4

    fresh_42

    Staff: Mentor

    ... isn't helpful. Why don't you write it as ##3^4(3^{4k+1})-5^2(5^{2k-1})## with the powers you have in your induction hypothesis? From here apply the formula I gave you and consider what you get on the right hand side.
     
  6. Mar 9, 2017 #5
    Okay, I got it.

    ##7|(9⋅3^{4k+1}-5⋅5^{2k-1})(14)-45(3^{4k+1}-5^{2k-1})##

    Thanks.
     
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