FIBONACCI NUMBERS ARE FASCINATING, AND THAT’S NO FIB! George Soliman Mathematics Instructor Raritan Valley Community College Two Truths and a Fib 1. The Fibonacci Sequence was discovered by Leonardo Fibonacci. FIB! th 2. The 6 Fibonacci number, F6, is the infinity symbol rotated 90°. TRUTH! ( = 8) 3. There exist explicit formulas for . 6 TRUTH! (stay tuned!) It All Began in Ancient India • The Fibonacci Sequence seemingly first appeared in the (the art of prosody) Chandaḥśāstra • Written by the ancient Indian mathematician and Sanskrit grammarian Pingala sometime between 450-200 BC • Describes the rhythm, stress, and intonation of speech, an important ancient Indian ritual Leonardo (6 Centuries Later) Fibonacci • c. 1170 – c. 1250 • Known as Leonardo Pisano or simply Leonardo of Pisa, since he was from Pisa • Given the name “Fibonacci”, which is a contraction of “Filius Bonacci” (Latin for “son of Bonacci”) in 1838 by the Franco-Italian historian Guillaume Libri • The Fibonacci Sequence was given its name in May 1876 by the French number theorist François Édouard Anatole Lucas Hindu-Arabic > Roman • Around 1190, Leonardo traveled to Bugia, Algeria (current-day Béjaïa) with his father, where he learned the Hindu-Arabic numeral system and computation methods (0-9, place value) • Up until then, Europeans were using Roman Numerals for computations (tally system for recording numbers) • He traveled extensively throughout northern Africa and the Middle East around the Mediterranean coast to study the various arithmetic systems then being used Liber Abaci • Around 1200, he returned home to Pisa • He realized the many advantages and conveniences of the Hindu-Arabic numeral system over Roman Numerals then being used in Italy • In 1202, Leonardo published his pioneering book entitled Liber Abaci (Latin for “The Book of Calculation”), which introduced the Hindu-Arabic numeral system and arithmetic methods to Europe • The book was so influential that by the end of the 16th century, most of Europe had adjusted to the current Hindu-Arabic system • He eventually wrote three more books Immortal Rabbits: Problem • One of the many arithmetic problems in Leonardo’s Liber Abaci is the “Problem of the Rabbits”: • “A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.” (Sigler 404) Immortal Rabbits: Solution • Each newborn pair of (immortal) beginning 1 rabbits, a male and a female, matures in first 2 one month and then starts to breed second 3 • “You can indeed see in the margin how third 5 we operated, namely that we added the first number to the second, namely the fourth 8 1 to the 2, and the second to the third, fifth 13 and the third to the fourth, and the sixth 21 fourth to the fifth, and thus one after another until we added the tenth to the seventh 34 eleventh, namely the 144 to the 233, eighth 55 and we had the abovewritten sum of rabbits, namely 377, and thus you can ninth 89 in order find it for an unending number tenth 144 of months.” (Sigler 404-405) eleventh 233 end 377 Immortal Rabbits: Closer Look • Let r represent a pair of Month Population Adults Newborns Total newborn rabbits Beginning R 1 0 1 1 Rr 1 1 2 2 RrR 2 1 3 • Let R represent a pair of 3 RrRRr 3 2 5 adult (“mature”) rabbits 4 RrRRrRrR 5 3 8 5 RrRRrRrRRrRRr 8 5 13 ⋮ ⋮ ⋮ ⋮ ⋮ Recursive Definition and the First 30 • = 1 1 1 11 89 21 10946 1 2 1 12 144 22 17711 • = 1 3 2 13 233 23 28657 4 3 14 377 24 46368 2 5 5 15 610 25 75025 • = + , 3 6 8 16 987 26 121393 7 13 17 1597 27 196418 −1 −2 ≥ 8 21 18 2584 28 317811 • Some define = 0 9 34 19 4181 29 514229 10 55 20 6765 30 832040 0 Fibonacci Spiral SOME COOL PROPERTIES! The good stuff! Sum of First n Fibonacci Numbers = : 1 = 1 1 1 11 89 21 10946 = : 1 + 1 = 2 2 1 12 144 22 17711 3 2 13 233 23 28657 = : 1 + 1 + 2 = 4 4 3 14 377 24 46368 5 5 15 610 25 75025 = : 1 + 1 + 2 + 3 = 7 6 8 16 987 26 121393 = : 1 + 1 + 2 + 3 + 5 = 12 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 General Formula: = ⋮ 10 55 20 6765 30 832040 � + − = Proof (Mathematical Induction) General Formula: = 1 +1 = + � +2 − � � +1 =1 =1 =1 = : = 1 = 1 = 2 1 = 1 = 1 + 1 3 − − +2 − +1 = Suppose that = 1 + − +2 � − QED =1 for some integer 1. ≥ Sum of First n Even Indices = : 1 = 1 1 1 11 89 21 10946 = : 1 + 3 = 4 2 1 12 144 22 17711 3 2 13 233 23 28657 = : 1 + 3 + 8 = 12 4 3 14 377 24 46368 5 5 15 610 25 75025 = : 1 + 3 + 8 + 21 = 33 6 8 16 987 26 121393 = : 1 + 3 + 8 + 21 + 55 = 88 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 General Formula: = ⋮ 10 55 20 6765 30 832040 � + − = Sum of First n Odd Indices = : 1 = 1 1 1 11 89 21 10946 = : 1 + 2 = 3 2 1 12 144 22 17711 3 2 13 233 23 28657 = : 1 + 2 + 5 = 8 4 3 14 377 24 46368 5 5 15 610 25 75025 = : 1 + 2 + 5 + 13 = 21 6 8 16 987 26 121393 = : 1 + 2 + 5 + 13 + 34 = 55 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 General Formula: = ⋮ 10 55 20 6765 30 832040 � − = Sum of Squares of Consecutive Fibonacci Numbers + = : + = 1 + 1 = 2 2 2 1 1 2 12 1 1 2 +1 = : + = 1 + 4 = 5 2 1 2 1 2 2 2 3 3 2 3 4 = : + = 4 + 9 = 13 4 3 4 9 2 2 3 4 5 5 5 25 = : + = 9 + 25 = 34 2 2 6 8 6 64 4 5 = : + = 25 + 64 = 89 7 13 7 169 2 2 8 21 8 441 5 6 General Formula: + = 9 34 9 1156 ⋮ + 10 55 10 3025 + + ∀ ∈ ℤ 11 89 11 7921 Sum of Squares of First n Fibonacci Numbers = : 1 = 1 1 1 1 1 = : 1 + 1 = 2 2 1 2 1 3 2 3 4 = : 1 + 1 + 4 = 6 4 3 4 9 5 5 5 25 = : 1 + 1 + 4 + 9 = 15 6 8 6 64 = : 1 + 1 + 4 + 9 + 25 = 40 7 13 7 169 8 21 8 441 9 34 9 1156 General Formula: = ⋮ 10 55 10 3025 � + 11 89 11 7921 = Why? Check Out Fibonacci Squares! Proof (Mathematical Induction) General Formula: = +1 = + 2 2 2 2 � +1 � � +1 =1 =1 =1 = : = 1 = 1 = = 1 1 = 1 = + 2 2 2 1 1 2 +1 +1 � = + Suppose that = +1 +1 2 = � +1 =1 ++ for some integer 1. QED ≥ Fibonacci Numbers and Multiples Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 2 1 12 144 22 17711 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 Every 6th Fibonacci Number exclusively is a multiple of 8 9 34 19 4181 29 514229 10 55 20 6765 30 832040 Fibonacci Numbers and Multiples Every 3rd Fibonacci Number exclusively is a multiple of 2 (even) 1 1 11 89 21 10946 2 1 12 144 22 17711 Every 4th Fibonacci Number exclusively is a multiple of 3 3 2 13 233 23 28657 4 3 14 377 24 46368 5 5 15 610 25 75025 Every 5th Fibonacci Number exclusively is a multiple of 5 6 8 16 987 26 121393 7 13 17 1597 27 196418 8 21 18 2584 28 317811 Every 6th Fibonacci Number exclusively is a multiple of 8 9 34 19 4181 29 514229 10 55 20 6765 30 832040 th Every Fibonacci Number exclusively is a multiple of Fibonacci Numbers and Divisibility 377 = = 29 13 14 1 1 11 89 21 10946 7 10946 2 1 12 144 22 17711 = = 842 13 3 2 13 233 23 28657 21 7 4 3 14 377 24 46368 317811 = = 24447 5 5 15 610 25 75025 13 28 6 8 16 987 26 121393 7 317811 7 13 17 1597 27 196418 = = 843 377 8 21 18 2584 28 317811 28 9 34 19 4181 29 514229 14 | iff | 10 55 20 6765 30 832040 (read divides if and only if divides ) Fibonacci Numbers and Divisibility • In case you’re curious… • If | , then: = + + + + 2 ⁄ +1− −1+1−2 −1 +1−3 ⋯ −1 1 Fibonacci Numbers and Primes If is prime, then is prime, except = 3 4 1 1 11 89 21 10946 Converse NOT true! = 4181 = 113 37 2 1 12 144 22 17711 19 � Contrapositive: If is composite, then is 3 2 13 233 23 28657 composite, except = 3 4 3 14 377 24 46368 4 5 5 15 610 25 75025 We know from Discrete Mathematics that 6 8 16 987 26 121393 there are infinitely many prime numbers, but it is still unknown whether there are infinitely 7 13 17 1597 27 196418 many prime Fibonacci Numbers 8 21 18 2584 28 317811 9 34 19 4181 29 514229 As of March 2017, there are only 34 known 10 55 20 6765 30 832040 prime Fibonacci Numbers, the largest of which is (21,925 digits!) 104911 Fibonacci Numbers and Primes Every prime divides a Fibonacci Number 1 1 11 89 21 10946 If a prime number ±1 mod 10 2 1 12 144 22 17711 (i.e.