Lifting the Exponent (LTE)

Explanations 7

• Lifting the Exponent (LTE) is a powerful tool in number theory. Before we get started, let's make sure we know a bit of notation: Definition: νp(x)\nu_p (x)νp​(x) denotes the la…

  By Vincent Huang, Po-Shen Loh, Yang Liu, Amir Hossein Parvardi, amaterasu, Christopher Liu, Vee, Victor Wang

• Applications of LTE Lifting the Exponent Lemma explains what LTE is, and its proof. The motivation should be quite obvious, and now we will see some nice applications of this po…

  By amaterasu

• a) Let xxx and yyy be two positive real numbers such that for each positive integer nnn, the number xn−ynx^n-y^nxn−yn is a positive integer. Show that xxx and yyy are both posit…

  By Amir Hossein Parvardi

• Let nnn be a square-free number. Show that there does not exist positive integers xxx and yyy such that (x+y)3∣xn+yn(x+y)^3 | x^n+y^n(x+y)3∣xn+yn.

  By Amir Hossein Parvardi

• (Balkan 1993). Let ppp be a prime number and m>1m>1m>1 be a positive integer. Show that if for some positive integers x>1,y>1x>1, y>1x>1,y>1 we have xp+yp2=(x+y2)m,\frac{x^p+y^p…

  By Amir Hossein Parvardi

• For some positive integer nnn, the number 3n−2n3^n-2^n3n−2n is a perfect power of a prime. Prove that nnn is a prime.

  By Amir Hossein Parvardi

• Confirm that, 3n+1∣∣23n+13^{n+1}||2^{3^n}+13n+1∣∣23n+1, or ν3(23n+1)=n+1\nu_3(2^{3^n}+1)=n+1ν3​(23n+1)=n+1. Note, this also means that there are infinite nnn such that n∣2n+1n|2…

  By amaterasu