Proof Without Words: Perfect Numbers and Triangular Numbers Roger B. Nelsen ([email protected]), Lewis & Clark College, Portland, OR Theorem ([1]). Every even perfect number, N p = 2 p−1 (2 p − 1) with p ≥ 3 prime, satisfies N p = 1 + 9Tn , where n = (2 p − 2)/3 and Tn is the nth triangular number. Proof. (Shown for p = 5 where N5 = 16 · 31 and n = 10.)

2 p–1 –1

2 p –1

2p–1

2 p–1

2 p –1

Np = 2p –1(2 p –1)

N p = T2 p –1 = T3n+1

T 3n+1 = 1 + 9Tn

Note that for p odd, 2 p − 2 ≡ (−1) p + 1 ≡ 0 (mod 3). The theorem holds for all odd p ≥ 3, although N p is not perfect for composite p. Summary. We show wordlessly that every even perfect number greater than six is one more than nine times a triangular number. References 1. C. F. Eaton, B. Kotkowski, Problem 1482, Perfect numbers in terms of triangular numbers, Math. Mag. 68 (1995) 307, http://dx.doi.org/10.2307/2690587, and 69 (1996) 308–309, http:// dx.doi.org/10.2307/2690542. http://dx.doi.org/10.4169/college.math.j.47.3.171 MSC: 11A25, 05A15

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