Author Archives: mzspivey

Proof of the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states the following: Every integer greater than 1 can be represented uniquely as the product of prime numbers. Another way to put this is that every integer has a unique factorization.  For example, 60 factors … Continue reading

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Equivalence of Two Binomial Sums

This month’s post entails proving the following equivalence: Identity 1: Despite the fact that these two sums are over the same range and are equal for all values of n they are not equal term-by-term. If you think about the two sides … Continue reading

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A Sum of Ratios of Binomial Coefficients

In this post we evaluate the sum .  Then we’ll generalize it and evaluate . The key tools we need for the first sum are the trinomial revision identity, , and the parallel summation identity, .  Using trinomial revision, we have . … Continue reading

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An Alternating Convolution Identity via Sign-Reversing Involutions

This month’s post is on the combinatorial proof technique of sign-reversing involutions.  It’s a really clever idea that can often be applied to identities that feature alternating sums.  We’ll illustrate the technique on the following identity. . (Here, we use … Continue reading

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A Mathematical Riddle

This past weekend I attended the wedding of a former student, Jake Linenthal.  On the dinner tables at the reception were sheets of paper containing a Mad Libs-style story of how Jake and his wife Abby got engaged, as well … Continue reading

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Infinite Series Expressions for Pi, E, Phi, and Gamma

Recently I was trying to find infinite series expressions for some famous mathematical constants, and I thought I would record what I found here.  This post considers , Euler’s constant e, the golden ratio , and the Euler-Mascheroni constant . … Continue reading

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Minimum Number of Wagons to Connect Every City in Ticket to Ride – Europe

How many wagons would you need to create a route network in the board game Ticket to Ride – Europe so that every city is connected to every other city through the network?  If you could do this you could potentially … Continue reading

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