Number Theory
http://www.maa.org/taxonomy/term/40567/0
enFetching Water with Least Residues
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/fetching-water-with-least-residues
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In a classic pouring problem, given two unmarked jugs with capacities \(m\) and \(n\) pints, where \(m\) and \(n\) are relatively prime integers, and an unlimited supply of water, the goal is to obtain exactly \( p\) pints, where \( p\) is an integer, \( 0 < p < m+n \). This capsule uses properties of least residues to show that there are two distinct pouring sequences to achieve the desired result. The more efficient sequence can be determined by solving a linear congruence.</em></p>
</div></div></div>Fermat's Last Theorem for Fractional and Irrational Exponents
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/fermats-last-theorem-for-fractional-and-irrational-exponents
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Following his senior seminars, the author discusses Fermat's last theorem for rational and irrational exponents, in which the rational solutions are characterized.</em></p>
</div></div></div>Combinations and Permutations
http://www.maa.org/programs/faculty-and-departments/course-communities/combinations-and-permutations
A Quick Change of Base Algorithm for Fractions
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-quick-change-of-base-algorithm-for-fractions
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This note is on the digital (floating-point) representation in various arithmetic bases of the reciprocal of an integer \( m \). An algorithm is given to change the representation of \( 1/m \) in base \( b \) to its representation in base \( b+mt \) for any integer \( t\).</em></p>
</div></div></div>Sam Loyd's Courier Problem with Diophantus, Pythagoras, and Martin Gardner
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/sam-loyds-courier-problem-with-diophantus-pythagoras-and-martin-gardner
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In Sam Loyd's classical Courier Problem, a courier goes around an army while both travel at constant speeds. If the army travels its length during the time the courier makes his trip, how far does the courier ride? In both revisions of this problem, a single-file army and a square army, the solution is irrational. Here, variations are considered in which the solutions are rational. In fact, certain Pythagorean tripes can be used to generate problems that have integer solutions.</em></p>
</div></div></div>Averaging Sums of Powers of Integers
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/averaging-sums-of-powers-of-integers
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>When is the average of sums of powers of integers itself a sum of the ﬁrst \(n\) integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions regarding when larger averages may have solutions.</em></p>
</div></div></div>Magic "Squares" Indeed!
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/magic-squares-indeed
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>A real matrix is called square-palindromic if, for every base \(b\), the sum of the squares of its rows, columns, and four sets of diagonals (as described in the article) are unchanged when the numbers are read "backwards" in base \(b\). The authors prove that all \(3 \times 3\) magic squares are square-palindromic. They also give sufficient conditions for \(n \times n\) magic squares to be square-palindromic, which include all circulant matrices and all symmetrical magic squares.</em></p>
</div></div></div>On Rational Function Approximations to Square Roots
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-rational-function-approximations-to-square-roots
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Using a result on periodic continued fractions, the author presents a rational function method of approximating square roots that is faster than Newton's method.</em></p>
</div></div></div>Fermat's Last Theorem for Gaussian Integer Exponents
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/fermats-last-theorem-for-gaussian-integer-exponents
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Andrew Wiles proves that Fermat's Last Theorem is false for integer exponents larger than \(2 \). Using the Gelfond-Schneider Theorem on transcendental numbers, the author generalizes Wiles' result easily by showing that Fermat's Last Theorem is false for Gaussian integer exponents that are not real.</p>
</div></div></div>Writing Numbers in Base \(3\), the Hard Way
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/writing-numbers-in-base-3-the-hard-way
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Every integer can be expressed in base \(2\) using the set \(\{-1, 0, 1\}\) as coefficients. Does one need to use this set, or might another set of numbers do as well? The author investigates this type of question in base \(3\), to provide richer examples.</em></p>
</div></div></div>Summing Cubes by Counting Rectangles
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/summing-cubes-by-counting-rectangles
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Starting with a homework problem on combinations, the capsule applies the "checkerboard" logic to derive identities involving summing squares and cubes. </em></p>
</div></div></div>Streaks and Generalized Fibonacci Sequences
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/streaks-and-generalized-fibonacci-sequences-0
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This capsule discusses an alternative way of examining the Fibonacci sequence. As a result, a class of generalized Fibonacci sequences of numbers can be defined.</em></p>
</div></div></div>No Arthmetic Cyclic Quadrilaterals
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/no-arthmetic-cyclic-quadrilaterals
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Based on the notion of "arithmetic triangles," arithmetic quadrilaterals are defined. It was proved by using an elliptic curve argument that no such quadrilateral can be inscribed on a circle. This capsule provides an elementary proof following the classic ideas of Euler and Fermat.</em></p>
</div></div></div>Searching for Möbius
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/searching-for-m-bius
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This capsule introduces a way of producing the Möbius function in Number Theory through the algebra of formal series. The author hopes that this exposition can lobby more mathematicians to integrate some formal algebra and combinatorics into the instruction of elementary number theory.</em></p>
</div></div></div>Parity and Primality of Catalan Numbers
http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/parity-and-primality-of-catalan-numbers
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Based on a closed form and recurrence relation of Catalan numbers, the authors proves their parity and primality. </em></p>
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