• 1 are expressible as the sum of two squares in exactly one way (e.g. 17 = 1^2 +
  42 ), while those of the form 4k + 3 (like 19) cannot be written as the sum of two
  squares at all. Joseph Lagrange also proved a famous mathematical theorem
  about square powers: every positive whole number is the sum of four squares.
  So, for example, 19 = 1^2 + 1^2 + 1^2 + 4^2. Higher powers have been explored
  and books filled with theorems, but many problems remain.
  We described the prime numbers as the ‘atoms of mathematics’. But ‘surely,’
  you might say, ‘physicists have gone beyond atoms to even more fundamental
  units, like quarks. Has mathematics stood still?’ If we limit ourselves to the
  counting numbers, 5 is a prime number and will always be so. But Gauss made a
  far-reaching discovery, that for some primes, like 5, 5 = (1 – 2i) × (1 + 2i)
  where of the imaginary number system. As the product of two Gaussian
  integers, 5 and numbers like it are not as unbreakable as was once supposed.

the condensed idea

The atoms of mathematics