I will be regularly updating this post with results, along with my proofs of them, that I encounter about multiplicative functions, in particular involving the Dirichlet product. A multiplicative function is such that for , , where . The Dirichlet product of two arithmetical functions (functions that take operate in the integer domain)  is . Result 1:…

# The Coin Problem

An interesting problem is the following: In a land with only denominations of \$6 and \$11, what is the largest integral expense that cannot be paid. Examples of expenses that cannot be paid in this strange land are 1, 2, 3, 4, 5, 7, 8, 9, 10. However, how do we find the largest such instance? It…

# Minkowski to the Rescue

I had been struggling for several days last year to find a proof for the intriguing Fermat’s Two Square theorem which states the following : If p is a prime number congruent to 1 mod 4, then it can be expressed as the sum of two squares. At about the same time I had also…

# Kummer’s Theorem on binomial coefficients

The Kummer’s theorem is a beautiful result concerning the divisibility of binomial coefficients(and so it is in a sense a connection between counting and number theory!). Binomial coefficients pop up everywhere. This may be because counting is one of those operations that human instinct can so very easily grasp. Though a binomial coefficient can have…

# Number Theory : The Queen of Mathematics

The Prince of Mathematics, Carl Friedrich Gauss once famously said : Mathematics is the Queen of the Sciences and Number theory is the Queen of Mathematics. This statement though highly debatable, does find resonance in many of the great mathematicians of both the past and the present (and most probably this will be true of…