Finding 8 Numbers with phi(n) = 240

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SUMMARY

The discussion focuses on finding eight distinct integers n such that the Euler's totient function φ(n) equals 240. The participants explore the mathematical properties of φ(n), specifically using the formula φ(n) = n * Π(1 - 1/p) for prime factors p of n. An example calculation is provided using n = 12^12, demonstrating how to derive φ(12^12) = 2^24 * 3^12, which is a crucial step in identifying suitable candidates for n.

PREREQUISITES
  • Understanding of Euler's totient function φ(n)
  • Familiarity with prime factorization
  • Knowledge of multiplicative functions in number theory
  • Basic algebraic manipulation skills
NEXT STEPS
  • Research the properties of Euler's totient function φ(n) in-depth
  • Learn about prime factorization techniques for large integers
  • Study multiplicative functions and their applications in number theory
  • Explore algorithms for finding integers with specific properties related to φ(n)
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Mathematicians, number theorists, and students interested in advanced topics related to the Euler's totient function and its applications in finding integers with specific properties.

AH05
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Find eight different numbers n with phi(n) = 240.



phi(n) = the number of positive integers less than or equal to n, that are relatively prime to n.
 
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Surely you are aware that you must show what you have tried yourself before anyone can give suggestions?
 
12^12 = (3^12)(2^24)

phi(12^12) = phi(3^12) * phi(2^24) = 2^(24-1) * (2-1) *3^12 *(3-1)
= 2^23 * 3^12 * 2 = 2^24 * 3^12
 

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