The Erdős–Ginzberg–Ziv theorem with units Original Research Article

Let image be a sequence of elements of image, the integers modulo image. How large must image be to guarantee the existence of a subsequence image and units image with image? Our main aim in this paper is to show that image is large enough, where image is the sum of the exponents of primes in the prime factorisation of image. This result, which is best possible, could be viewed as a unit version of the Erdős–Ginzberg–Ziv theorem. This proves a conjecture of Adhikari, Chen, Friedlander, Konyagin and Pappalardi. We also discuss a number of related questions, and make conjectures which would greatly extend a theorem of Gao.