triangular number theorem

n:Nat   |   n>0   |--   2(n-1)(2n-1) = 1 + 2 + 3 + ... + (2n-1)

proof: by induction

base case n = 1

2^(1-1)(2¹-1) = 1(2-1) = 1 = 2¹-1

inductive step: assume true for n = k; prove for n = k+1

1 + 2 + 3 + ... + (2^(k+1)-1)

= 1 + 2 + 3 + ... (2^k-1) + 2^k + (2^k+1) ... + (2^(k+1)-1)

= 1 + 2 + 3 + ... (2^k-1) + 2^k*2^k + 1 + 2 + ... + (2^k-1)

= 2(1 + 2 + 3 + ... (2^k-1)) + 2^k*2^k

= 2*2^(k-1)(2^k-1) + 2^k*2^k

= 2^k(2^k-1 + 2^k)

= 2^k(2^(k+1)-1)

• If 2ⁿ-1 is a Mersenne prime, then 2^(n-1)(2ⁿ-1) is an even perfect number. So all even perfect numbers are triangular.