Abstract
When can we express a prime number as a sum of two squares? Let’s start by sorting the first dozen primes into those with such an expression, and the rest:
Do you see a pattern?
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Notes
- 1.
Fermat’s Last Theorem was for 350 years in fact just a conjecture: the equation \({x}^{n} + {y}^{n} = {z}^{n}\) has no integer solutions when n ≥ 3. The proof of this conjecture, found only in the mid-1990’s, is one of the great achievements of twentieth-century mathematics.
- 2.
This name is slightly nonstandard; the literature usually refers to \(\mathbb{Z}[i]\) as the ring of Gaussian integers. I prefer to drop the -ian ending, as it is unnecessary, inconsistently applied, and awkward to pronounce with some non-English names.
- 3.
Here π stands for a Gauss integer, not 3.141592….
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Trifković, M. (2013). Examples. In: Algebraic Theory of Quadratic Numbers. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7717-4_1
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DOI: https://doi.org/10.1007/978-1-4614-7717-4_1
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