Testing Math

I am a mathematician, so it is befitting I speak on it. My primary area is Number Theory, specifically supersingular elliptic curves and Post-Quantum Cryptography. I also integrate Probability Theory with Number Theory and study random properties of arithmetic.

I also get to use this as an opportunity to test out MathJax integration in Jekyll for making my website. For the programmers, I added the block

 <!-- MathJax configuration -->
<script>
  window.MathJax = {
    tex: {
      inlineMath: [['\\(', '\\)'], ['$', '$']],
      displayMath: [['\\[', '\\]'], ['$$', '$$']]
    },
    options: {
      skipHtmlTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
    }
  };
</script>

to my _site/index.html file in my website repo.

Patch note: that didn’t actually work. GitHub Pages regenerates _site/ from source on every push, so edits there are thrown away. The configuration (plus the script tag that loads MathJax) now lives in _includes/mathjax.html, which the theme’s head.html includes on every page.

Now I will present a fun fact.

Number Theory

Fact: A prime \(p \ge 3\) can be written as a sum of two squares if and only if \(p \equiv 1 \pmod{4}\).

Proof: \((\Rightarrow)\) Suppose \(p=x^2+y^2\). Computation mod 4 reveals \(z^2 \equiv 0, 1 \pmod{4}\) for all integers \(z\). Then \(p \equiv x^2 + y^2 \pmod{4}\) is congruent to 0, 1, or 2 mod 4. As \(p\) is an odd prime, \(p\) must be 1 mod 4.

\((\Leftarrow)\) Suppose \(p \equiv 1 \pmod{4}\) is prime. As \(p\) is an odd prime, \(p\) is not ramified in the Gaussian integers \(\mathbb{Z}[i]\). As \(p \equiv 1 \pmod{4}\), the prime \(p\) splits in \(\mathbb{Z}[i]\). Since \(\mathbb{Z}[i]\) is a principal ideal domain, the primes above \(p\) are principal, say \((a+ib)\) and \((a-ib)\), so \(p = (a+ib)(a-ib) = a^2 + b^2\) up to a unit (which we can absorb into \(a+ib\)). \(\square\)