@MISC{NAT_5.MIZ,
  AUTHOR = {Riccardi, Marco},
  TITLE = {The Perfect Number Theorem and {W}ilson's Theorem},
  SECTION1 = {Preliminaries},
  SECTION2 = {Wilson's Theorem},
  SECTION3 = {All Primes Congruent to 1 Modulo 4 are the Sum of Two Squares},
  SECTION4 = {The Sum of Divisors Function},
  SECTION5 = {Two Basic Theorems on Perfect Numbers},
  SECTION6 = {A Formula Involving Euler's $\phi$ Function},
  DAY = {03},
  MONTH = {March},
  YEAR = {2009},
  ADDRESS1 = {Casella Postale 49\\ 54038 Montignoso, Italy},
  SUMMARY = {This article formalizes proofs of some elementary 
  theorems of number theory (see \cite{PFTB, LeVeque}): 
  Wilson's theorem (that $n$ is prime iff $n > 1$ and $(n-1)! \cong -1$ (mod $n$)),
   that all primes (1 mod 4) equal the sum of two squares, 
  and two basic theorems of Euclid and Euler about perfect numbers. 
  The article also formally defines Euler's sum of divisors 
  function $\phi$, proves that $\phi$ is multiplicative 
  and that $\sum_{k|n} \phi (k) = n$.}}

