Saidak, Filip A new proof of Euclid’s theorem. (English) Zbl 1228.11011 Am. Math. Mon. 113, No. 10, 937-938 (2006). The author provides a new proof of Euclid’s theorem that the number of primes is infinite, as the result of a process that may be continued indefinitely: Let \(N_1>1\). Then \(N_1\) and \(N_1+1\) are coprime because they are consecutive integers, so \(N_2=N_1(N_1+1)\) has at least two distinct prime factors. Then as before, \(N_2\) and \(N_2+1\) are consecutive integers, so \(N_3=N_2(N_2+1)\) must have at least three distinct prime factors, and so on. Reviewer: Olaf Ninnemann (Berlin) Cited in 2 ReviewsCited in 4 Documents MSC: 11A41 Primes Keywords:Euclid theorem; new proof PDFBibTeX XMLCite \textit{F. Saidak}, Am. Math. Mon. 113, No. 10, 937--938 (2006; Zbl 1228.11011) Full Text: DOI Online Encyclopedia of Integer Sequences: Sylvester’s sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2. a(n) = a(n-1)^2 + a(n-1), a(0)=1. List of primes generated by factoring successive integers in Sylvester’s sequence (A000058). Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester’s sequence). Largest prime factor of A000058(n) = A007018(n) + 1 (Sylvester’s sequence).