On a conjecture involving Fermat's Little Theorem

Author

John Clark, University of South Florida

Graduation Year

2008

Document Type

Thesis

Degree

M.A.

Degree Granting Department

Mathematics and Statistics

Major Professor

Stephen Suen, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Committee Member

Arthur Danielyan, Ph.D.

Keywords

Number Theory, Prime Numbers, Divisibility, Congruences, Sums of Powers of Consecutive Integers

Abstract

Using Fermat’s Little Theorem, it can be shown that Σ^m_i=1 i ^m−1 ≡ −1 (mod m) if m is prime. It has been conjectured that the converse is true as well. Namely, that Σ^m_i=1 i ^m−1 ≡ −1 (mod m) only if m is prime. We shall present some necessary and sufficient conditions for the conjecture to hold, and we will demonstrate that no counterexample exists for m ≤ 10¹² .

Scholar Commons Citation

Clark, John, "On a conjecture involving Fermat's Little Theorem" (2008). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/178