The Bernoulli Numbers Savage March 16, 2021 Outline 1 Sm(n) and its History 2 The Bernoulli Method 3 Euler’s Impact and Applications Savage The Bernoulli Numbers Chapter 1 Sm(n) and its History Savage The Bernoulli Numbers Sums of Powers of Integers Definition Let n; m 2 N with n ≥ 1. The sums of powers of integers, denoted Sm(n), is n X m m m m m Sm(n) := k = 1 + 2 + 3 + ··· + n : k=1 *Historically, finding quick ways of computing these sums were initially intriguing. *Applications in physics, engineering, and possibly in scripts for games such as Dota 2. Savage The Bernoulli Numbers Sums of Powers of Integers Example (Forms you may have seen) If m = 0, then n X 0 0 0 0 S0(n) = k = 1 + 2 + ··· + n = n: k=1 If m = 1, then n X n + 1 S (n) = k1 = 1 + 2 + ··· + n = : 1 2 k=1 Savage The Bernoulli Numbers Historical Questions Big Question: Can we find a “nice” formula for the sums of powers of integers? That is, is there a concise way of computing n X m m m m m Sm(n) = k = 1 + 2 + 3 + ··· + n ? k=1 Savage The Bernoulli Numbers Historical Questions Example (Geometric Sums) Let m ≥ 1. Then for b > 0 with b 6= 1, we know that bm − 1 b0 + b1 + b2 + b3 + ··· + bm−1 = b − 1 Savage The Bernoulli Numbers Historical Questions Big Question: Can we find a “nice” formula for the sums of powers of integers? That is, is there a concise way of computing n X m m m m m Sm(n) := k = 1 + 2 + 3 + ··· + n ? k=1 We have know of nice formulas for m = 0 and m = 1, but what if m > 1? Is each formula related somehow or is each independent? Savage The Bernoulli Numbers Historical Questions Example (Related?) We know n + 1 S (n) = n and S (n) = ; 0 1 2 but are these formulas (and others) related somehow? Savage The Bernoulli Numbers History: Archimedes (287-212 BC) of Syracuse, Sicily Our story begins with Archimedes. He is credited with first discovering that n + 1 S (n) = : 1 2 It’s also believed Archimedes also discov- ered a nice formula for S2(n). Savage The Bernoulli Numbers History: Aryabhata (476-550) of Kusumapura?, India Aryabhata was an Indian astronomer. He is credited with discovering a nice formula for S3(n), namely 2 S3(n) = (1 + 2 + ··· + n) : n + 12 n2(n + 1)2 ) S (n) = = 3 2 4 Savage The Bernoulli Numbers History: Abu Ali al-Hassan ibn al Haytham (965-1039) of Basra, Iraq Commonly referred to as Alhazan. Ar- guably discovered that 1 1 1 1 S (n) = n5 − n4 + n3 − n: 4 5 2 3 30 Savage The Bernoulli Numbers History: Thomas Harriot (1560-1621) of Oxford, U.K. and Pierre De Fermat (1601-1665) of Beaumont-de-Lomagne, France Both independently “proved” 1 1 1 1 S (n) = n5 − n4 + n3 − n: 4 5 2 3 30 *Given Sm(n), Fermat developed a recur- sive formula using integration to compute Sm+1(n). Savage The Bernoulli Numbers History: Johann Faulhaber (1580-1635) of Ulm, Germany In 1610, found nice formulas for Sm(n) where 0 ≤ m ≤ 10: In 1631, he published in Academia Algebrae nice formulas for Sm(n) where 0 ≤ m ≤ 23: Savage The Bernoulli Numbers The Faulhaber Method n+1 Given n ≥ 1, let α = S1(n) = 2 . Theorem (Fact - Key Observation) If m ≥ 1 is odd, then Sm(n) can be expressed as rational polynomial in α. Savage The Bernoulli Numbers The Faulhaber Method Example n+1 Given n ≥ 1, let α = 2 . 1 m = 1 ) S1(n) = α. 2 2 2 m = 3 ) S3(n) = (1 + 2 + ··· + n) = α . 3 4 3 1 2 m = 5 ) S5(n) = 3 α − 3 α . 4 m = 13 ) 64 80 656 944 2764 691 S (n) = α7 − α6 + α5 − α4 + α3 − α2: 13 7 3 15 21 105 105 Savage The Bernoulli Numbers The Faulhaber Method Example (Continued) Suppose m = 13 and n = 4. Then 13 13 13 13 S13(4) = 1 + 2 + 3 + 4 = 68; 711; 380 : 4+1 In this case, α = 2 = 10 and 64 80 656 944 2764 691 (10)7 − (10)6 + (10)5 − (10)4 + (10)3 − (10)2 7 3 15 21 105 105 is indeed 68; 711; 380 . Savage The Bernoulli Numbers Chapter 2 The Bernoulli Method Savage The Bernoulli Numbers Jacob Bernoulli (1655-1705) of Basel, Switzerland Effectively used Fermat’s methods to compute Sm(n) for 1 ≤ m ≤ 10: Savage The Bernoulli Numbers Bernoulli’s Sums, b = n − 1 1 2 1 1 S1(b) = 2 n − 2 n 1 3 1 2 1 1 S2(b) = 3 n − 2 n + 6 n 1 4 1 3 1 2 S3(b) = 4 n − 2 n + 4 n 1 5 1 4 1 3 1 1 S4(b) = 5 n − 2 n + 3 n − 30 n 1 6 1 5 5 4 1 2 S5(b) = 6 n − 2 n + 12 n − 12 n 1 7 1 6 1 5 1 3 1 1 S6(b) = 7 n − 2 n + 2 n − 6 n + 42 n 1 8 1 7 7 6 7 4 1 2 S7(b) = 8 n − 2 n + 12 n − 24 n + 12 n 1 9 1 8 2 7 7 5 2 3 1 1 S8(b) = 9 n − 2 n + 3 n − 15 n + 9 n − 30 n 1 10 1 9 3 8 7 6 1 4 3 2 S9(b) = 10 n − 2 n + 4 n − 10 n + 2 n − 20 n Savage The Bernoulli Numbers 1 h 2 1i S1(b) = 2 n − 1n 1 h 3 3 2 1 1i S2(b) = 3 n − 2 n + 2 n 1 h 4 3 2i S3(b) = 4 n − 2n + 1n 1 h 5 5 4 5 3 1 1i S4(b) = 5 n − 2 n + 3 n − 6 n 1 h 6 5 5 4 1 2i S5(b) = 6 n − 3n + 2 n − 2 n 1 h 7 7 6 7 5 7 3 1 1i S6(b) = 7 n − 2 n + 2 n − 6 n + 6 n 1 h 8 7 14 6 7 4 2 2i S7(b) = 8 n − 4n + 3 n − 3 n + 3 n 1 h 9 9 8 7 21 5 3 3 1i S8(b) = 9 n − 2 n + 6n − 5 n + 2n − 10 n 1 h 10 9 15 8 6 4 3 2i S9(b) = 10 n − 5n + 2 n − 7n + 5n − 2 n Savage The Bernoulli Numbers 1 h 2 −1 1i 2 n + (2)( 2 )n 1 h 3 −1 2 1 1i 3 n + (3)( 2 )n + (3)( 6 )n 1 h 4 −1 3 1 2i 4 n + (4)( 2 )n + (6)( 6 )n 1 h 5 −1 4 1 3 −1 1i 5 n + (5)( 2 )n + (10)( 6 )n + (5)( 30 )n 1 h 6 −1 5 1 4 −1 2i 6 n + (6)( 2 )n + (15)( 6 )n + (15)( 30 )n 1 h 7 −1 6 1 5 −1 3 1 1i 7 n + (7)( 2 )n + (21)( 6 )n + (35)( 30 )n + (7)( 42 )n 1 h 8 −1 7 1 6 −1 4 1 2i 8 n + (8)( 2 )n + (28)( 6 )n + (70)( 30 )n + (28)( 42 )n 1 h 9 −1 8 1 7 −1 5 1 3 −1 1i 9 n + (9)( 2 )n + (36)( 6 )n + (126)( 30 )n + (84)( 42 )n + (9)( 30 )n 1 h 10 −1 9 1 8 −1 6 1 4 −1 2i 10 n + (10)( 2 )n + (45)( 6 )n + (210)( 30 )n + (210)( 42 )n + (45)( 30 )n Savage The Bernoulli Numbers 1 h 2 −1 1i 2 1n + (2)( 2 )n 1 h 3 −1 2 1 1i 3 1n + (3)( 2 )n + (3)( 6 )n 1 h 4 −1 3 1 2i 4 1n + (4)( 2 )n + (6)( 6 )n 1 h 5 −1 4 1 3 −1 1i 5 1n + (5)( 2 )n + (10)( 6 )n + (5)( 30 )n 1 h 6 −1 5 1 4 −1 2i 6 1n + (6)( 2 )n + (15)( 6 )n + (15)( 30 )n 1 h 7 −1 6 1 5 −1 3 1 1i 7 1n + (7)( 2 )n + (21)( 6 )n + (35)( 30 )n + (7)( 42 )n 1 h 8 −1 7 1 6 −1 4 1 2i 8 1n + (8)( 2 )n + (28)( 6 )n + (70)( 30 )n + (28)( 42 )n 1 h 9 −1 8 1 7 −1 5 1 3 −1 1i 9 1n + (9)( 2 )n + (36)( 6 )n + (126)( 30 )n + (84)( 42 )n + (9)( 30 )n 1 h 10 −1 9 1 8 −1 6 1 4 −1 2i 10 1n + (10)( 2 )n + (45)( 6 )n + (210)( 30 )n + (210)( 42 )n + (45)( 30 )n Savage The Bernoulli Numbers First six rows Row # n1 n2 n3 n4 n5 n6 n7 (1) 2 1 (2) 3 3 1 (3) 6 4 1 (4) 5 10 5 1 (5) 15 15 6 1 (6) 7 35 21 7 1 *Looks like Pascal’s triangle! *(1655) Savage The Bernoulli Numbers First six rows completed Row # n1 n2 n3 n4 n5 n6 n7 (1) 2 1 (2) 3 3 1 (3) 4 6 4 1 (4) 5 10 10 5 1 (5) 6 15 20 15 6 1 (6) 7 21 35 35 21 7 1 Savage The Bernoulli Numbers Bernoulli Sums with Pascal numbers 1 h 2 −1 1i 2 1n + (2)( 2 )n 1 h 3 −1 2 1 1i 3 1n + (3)( 2 )n + (3)( 6 )n 1 h 4 −1 3 1 2 1i 4 1n + (4)( 2 )n + (6)( 6 )n + (4)(0)n 1 h 5 −1 4 1 3 2 −1 1i 5 1n + (5)( 2 )n + (10)( 6 )n + (10)(0)n + (5)( 30 )n 1 h 6 −1 5 1 4 3 −1 2 1i 6 1n + (6)( 2 )n + (15)( 6 )n + (20)(0)n + (15)( 30 )n + (6)(0)n 1 h 7 −1 6 1 5 4 −1 3 2 1 1i 7 1n + (7)( 2 )n + (21)( 6 )n + (35)(0)n + (35)( 30 )n + (21)(0)n + (7)( 42 )n 1 h 8 −1 7 1 6 5 −1 4 3 1 2 1i 8 1n + (8)( 2 )n + (28)( 6 )n + (56)(0)n + (70)( 30 )n + (56)(0)n + (28)( 42 )n + (8)(0)n 1 h 9 −1 8 1 7 6 −1 5 4 1 3 2 −1 1i 9 1n + (9)( 2 )n + (36)( 6 )n + (84)(0)n + (126)( 30 )n + (126)(0)n + (84)( 42 )n + (36)(0)n + (9)( 30 )n 1 h 10 −1 9 1 8 7 −1 6 5 1 4 3 −1 2 1i 10 1n + (10)( 2 )n + (45)( 6 )n + (120)(0)n + (210)( 30 )n + (252)(0)n + (210)( 42 )n + (120)(0)n + (45)( 30 )n + (10)(0)n *Each red and yellow number is a Pascal number.