Proofs of the Infinitude of Prime Numbers Using the Euler Totient Function and the Mobius Function

Tabsum B

Abstract: What's particularly striking about this collection of proofs is how elegantly they extend Euclid?s ancient logic using more modern number-theoretic tools like the Euler?s totient and M?bius functions. Rather than simply reiterating the infinitude of primes, the author reshapes the argument with a fresh lens-employing parity-based reasoning and properties of coprimality to construct numbers that inevitably lead to contradiction if the list of primes were finite. This suggests that arithmetic functions-though defined based on known primes-can paradoxically reveal unknown ones. It is evident that this approach bridges classical and contemporary methods, offering a pedagogically rich and logically refined pathway to explore prime distribution. While some steps may benefit from tighter formalism, especially in transitions between cases, the underlying structure showcases a meaningful fusion of intuition and rigor. That said, the way parity, coprime constructs, and prime decomposition converge here opens up a thoughtful avenue for future inquiry-perhaps even toward generalizations using other multiplicative functions. One cannot help but wonder: could similar logic extend to primes in specific congruence classes or to more complex algebraic structures?

Keywords: Euler's Totient Function, Mobius Function, Coprimality, Infinitude of Primes, Parity argument

How to Cite?: Tabsum B, "Proofs of the Infinitude of Prime Numbers Using the Euler Totient Function and the Mobius Function", Volume 14 Issue 11, November 2025, International Journal of Science and Research (IJSR), Pages: 381-382, https://www.ijsr.net/getabstract.php?paperid=MR251106205742, DOI: https://dx.doi.org/10.21275/MR251106205742