Proving Prime p≥3 Satisfies pr ≡ 1, 5, 7 or 11 (mod 12)

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SUMMARY

The discussion focuses on proving that for any prime number p greater than 3, the expression pr is congruent to 1, 5, 7, or 11 modulo 12. Participants emphasize that primes greater than 3 can be expressed in the forms 6n+1 or 6n+5, which helps in determining their behavior under modulo 12. The conversation also suggests exploring the impossibility of other residues, such as 2 modulo 12, by analyzing the properties of odd numbers and their congruences. This leads to a definitive conclusion regarding the valid residues for pr modulo 12.

PREREQUISITES
  • Understanding of modular arithmetic, specifically modulo 12.
  • Familiarity with prime number properties, particularly forms of primes greater than 3.
  • Basic knowledge of congruences and their implications in number theory.
  • Ability to analyze odd and even integers in modular contexts.
NEXT STEPS
  • Study the properties of primes in modular arithmetic, focusing on primes greater than 3.
  • Learn about the implications of the Chinese Remainder Theorem in relation to modular equations.
  • Investigate the behavior of odd numbers under various moduli, particularly mod 12.
  • Explore advanced topics in number theory, such as quadratic residues and their applications.
USEFUL FOR

This discussion is beneficial for students of number theory, mathematicians interested in modular arithmetic, and anyone seeking to deepen their understanding of prime number properties in modular contexts.

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Homework Statement



If p is a prime and p>3, show that pr\equiv1,5,7 or 11 (mod12)

Homework Equations


The Attempt at a Solution


Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.
 
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Amannequin said:

Homework Statement



If p is a prime and p>3, show that pr\equiv1,5,7 or 11 (mod12)

Homework Equations


The Attempt at a Solution


Do I go about this by knowing that any prime p greater than 3 is of the form 6n+1 or 6n+5? Any direction on how to go about this will be helpful. Thanks.

Show any other possibility doesn't work. For example, suppose ##p^r\equiv 2 (mod 12)##. What's wrong with that?
 
If k is an odd number, list the possibilities for k mod 12.
Next if p is a prime > 3, list the possibilities for p mod 12.
Then what conclusion can you draw about ##p^r##?
 

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