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Full-Text Articles in Number Theory
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Rose-Hulman Undergraduate Mathematics Journal
We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern
Irreducibility And Galois Groups Of Random Polynomials, Hanson Hao, Eli Navarro, Henri Stern
Rose-Hulman Undergraduate Mathematics Journal
In 2015, I. Rivin introduced an effective method to bound the number of irreducible integral polynomials with fixed degree d and height at most N. In this paper, we give a brief summary of this result and discuss the precision of Rivin's arguments for special classes of polynomials. We also give elementary proofs of classic results on Galois groups of cubic trinomials.