27 citations · 81 across the 27 of their papers we have counts for
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Legendre-Teege Reciprocity
Mark B. Villarino
Legendre published the first attempted proof of the law of Quadratic Reciprocity. But, in its final form (1797), it had a gap. Some 125 years later Herman Teege published the first…
Archimedes' Revenge
Mark B. Villarino, Joseph C. Várilly, Runze Li
We offer an instructive solution to the problem of computing the volume of the orthogonal intersection of three hyperboloids.
Legendre's Singular Modulus
Mark B. Villarino
We elaborate Legendre's original proof of his discovery of the first singular modulus in the history of mathematics, as well as its appearance in Ramanujan's formula for the arc le…
Hilbert's Proof of His Irreducibility Theorem
Mark B. Villarino, Bill Gasarch, Kenneth Regan
Hilbert's Irreducibility Theorem is a cornerstone that joins areas of analysis and number theory. Both the genesis and genius of its proof involved combining real analysis and comb…
A Cubic Surface of Revolution
Mark B. Villarino
We develop a direct and elementary (calculus-free) exposition of the famous cubic surface of revolution x^3+y^3+z^3-3xyz=1.12 pages. We have added a second elementary proof that th…
Mertens' Proof of Mertens' Theorem
Mark B. Villarino
We present a self-contained elementary and detailed exposition of Mertens' own proof of his theorem on the divergence of the series of the reciprocals of the primes and compare it…
Ramanujan's Most Singular Modulus
Mark B. Villarino
We present an elementary self-contained detailed computation of Ramanujan's most famous singular modulus, k210, based on the Kronecker Limit Formula.