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New Looks at Old Number Theory

Aimeric Malter, Dierk Schleicher and Don Zagier
The American Mathematical Monthly
Vol. 120, No. 3 (March 2013), pp. 243-264
DOI: 10.4169/amer.math.monthly.120.03.243
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Page Count: 22
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New Looks at Old Number Theory
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Abstract We present three results of number theory that all have classical roots, but also modern aspects. We show how to (1) systematically count the rational numbers by iterating a simple function, (2) find a representation of any prime congruent to 1 modulo 4 as a sum of two squares by using simple properties of involutions and pairs of involutions, and (3) find counterexamples to Euler’s conjecture that a fourth power can never be the sum of three fourth powers by using properties of quadratic polynomials with rational coefficients.

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