ETY CI O O F S The Dozenal Society of America E 0 1 A L X 2 M A 9 3 E The Personality of the Integers N R E 8 4 I Z 7 5 C from One to One Gross A O 6 D Jay Schiffman S Introduction n his fascinating paperback entitled The Lore of integer that lends a special personality to them es- I Large Numbers,1 author Philip J. Davis devotes a pecially when one resorts to duodecimals. Finally, detailed chapter to The Personality of Numbers and my goal is to furnish number theoretic characteris- provides a list entitled Who’s Who Among The Inte- tics related to each of the integers that are perhaps gers from One to One Hundred in the decimal base. omitted from the average list. Our initial task is to The purpose of my paper is to expand his list to the present the reader with a dictionary which is termed first gross and present additional characteristics of A Glossary of Symbols and Terms. These symbols these integers. It is indeed the uniqueness of each and terms will be illustrated in this section. A Glossary of Symbols and Terms 1. n! The factorial of the positive integer n. We de- 6. Ln The nth Lucas Number. n is a Lucas Number fined n!, known as n factorial, recursively as if n satisfied the Lucas Sequence defined recur- follows: 0! = 1 and (n + 1)! = (n + 1)(n!). sively as follows: L1 = 1, L2 = 3, and Ln = Hence 1! = (0 + 1)(0!) = (1)(1) = 1 and Ln−2 + Ln−1 for n ≥ 3. Hence L3 = L1 + L2 5! = (4 + 1)(4!) = (5)(20) = X0. = 1 + 3 = 4. 2. Tn The nth Triangular Number. We defined 7. pn The nth Prime Number. Define a prime num- the nth Triangular Number Tn as follows: Tn ber to be any integer p > 1 that has only 1 and = 1 + 2 + 3 + ::: + n = ((n)(n + 1)=2. Thus T5 p as its factors (divisors). Hence p3 = 5; for the = 1 + 2 + 3 + 4 + 5 = 13 = ((5)(5 + 1))=2. only divisors of 5 are 1 and 5. 8. FEn The nth Fermat Prime. n is a Fermat Prime 3. Sn The nth Square Number. Define the nth n if n is prime and n = 22 +1 for any whole num- Square Number Sn by the formula Sn = 1 + 2 2 2 3 + 5 + 7 + ::: + (2n − 1) = n . Hence S4 ber n. To cite an example, FE2 = 2 + 1 = 15. = 1 + 3 + 5 + 7 = 14 = 42. 9. COn The nth Integer for which a regular n-gon 4. THn The nth Tetrahedral Number. The is constructible using solely a straight-edge and Tetrahedral Numbers form the sequence compass. This idea is related to the geometric 1; 4; X; 18; 2E; 48;:::. The Tetrahedral Numbers notion of Constructible Regular n-gons in the are related to the Triangular Numbers as fol- following sense as can be demonstrated via ab- n lows: THn = THn−1 + Tn for n ≥ 2 and TH1 stract algebraic techniques: The sole values of n = 1. Hence TH5 = TH4 + T5 = 18 + 13 = 2E. for which a regular -gon is constructible using only a straight-edge and compass are such that 5. Fn The nth Fibonacci Number. n is a Fibonacci the only odd primes dividing n are the Fermat Number if n satisfies the Fibonacci Sequence Primes whose squares do not divide n. Hence defined recursively as follows: F1 = 1, F2 = 1, a regular duodecahedron [sic] (10 sides) is con- 2 and Fn = Fn−2 + Fn−1 for n ≥ 3. Hence F3 = structible in the above sense; for 10 = (2 )(3) F1 + F2 = 1 + 1 = 2. and 3 is a Fermat Prime (F1 = 3 as the reader 1Philip J. Davis, The Lore of Large Numbers. The New Mathematical Library, New York, 1961. 1.