Factoring and Divisibility Problems with 2^n - 1: Beginner Proof Method

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SUMMARY

The discussion focuses on factoring the expression 215 - 1 and finding an integer x such that 1 < x < 232767 - 1, where 232767 is divisible by x. The factorization of 215 - 1 results in the product of 31 and 1023. For the second part, it is established that x must be a power of 2, as only powers of 2 can evenly divide another power of 2. The relationship between these two problems lies in the properties of divisibility and factorization of powers of 2.

PREREQUISITES
  • Understanding of factorization techniques, specifically for expressions of the form 2n - 1.
  • Knowledge of divisibility rules, particularly regarding powers of integers.
  • Familiarity with the concept of integer sequences and their properties.
  • Basic algebraic manipulation skills to handle polynomial expressions.
NEXT STEPS
  • Study the factorization of expressions of the form 2n - 1, including specific cases like 23 - 1 and 25 - 1.
  • Learn about the properties of powers of 2 and their divisibility.
  • Explore integer sequences and their applications in number theory.
  • Investigate polynomial factorization techniques, particularly for cubic and higher-degree polynomials.
USEFUL FOR

This discussion is beneficial for students studying number theory, mathematicians interested in divisibility and factorization, and educators looking for examples of problems involving powers of integers.

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Homework Statement



I'm given this problem and I think I'm supposed to use the same or similar method to solve both of its parts:

a) Factor 2^{15} - 1 = 32,767 into a product of two smaller positive integers.
b) Find an integer x such that 1 &lt; x &lt; 2^{32767} - 1 and 2^{32767} is divisible by x.

Homework Equations



It is shown above the problem that:
x = 1 * 2 * 3 * 4 * ... * (n + 1) + 2 = 2 * (1 * 3 * 4 * ... *(n + 1) + 1
While I get that it's true, I don't quite see how I can apply the same to solving the problem.Can anyone give a hint?

The Attempt at a Solution



I tried "guessing", however with no success.
 
Last edited:
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I don't see any connection between the two.
Can you factorise x3-1?
 
For b, you realize that 2^{32767} can be divided evenly only by another power of 2, right? So x must be a power of 2.
 

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