Wednesday, June 17, 2015

The Mathematics of Today

6/17/2015

Today is the 17th day of the month.  The properties of 17 include:
 17 is a prime number.  It is a Chen, emirp, good, hungry, Pierpont, and strong prime.  It is the smaller prime in a twin prime pair, and the larger prime in a cousin prime pair. 17 is an amenable, arithmetic, cyclic, deficient, evil, hungry, odd, pernicious, polite, Proth, and Woodall number. It has 2 divisors, whose sum is σ = 18. Its totient is φ = 16.  The product of its digits is 7, while the sum is 8. 17 can be expressed as the sum of two squares: 17 = 1^2 + 4^2.  It can also be express as the sum of two fourth powers: 17 = 1^4 + 2^4 (it is the smallest integer to have this property).  17 can also be expressed as the difference of two squares: 17 = 9^2 – 8^2.  17 is the hypotenuse of a Pythagorean triangle: 17^2 = 15^1 + 8^2. It can be written as a sum of positive squares in only one way, i.e., 17 = 16 + 1 = 4^2 + 1^2. 17 is nontrivially palindromic in base 2, and base 4. 17 is an esthetic number in base 4, base 5, base 7, base 8 and base 15, because in such bases it adjacent digits differ by 1. It is a Leyland number of the form 3^2 + 2^3. 17 is a palindromic number when written in base 2, base 4, and base 16.  17 is an undulating number when written in base 4. It is the 10-th Perrin number. It is a plaindrome in base 3, base 6, base 7, base 9, base 10, base 11, base 12, base 13, base 14, base 15 and base 16.  It is a nialpdrome in base 5, base 8 and base 16.  It is a zygodrome in base 16. A polygon with 17 sides can be constructed with ruler and compass. 17 is an equidigital number, since it uses as much as digits as its factorization. There are 17 distinct sets of regular polygons that can be packed around a point (e.g., 4 squares, 2 hexagons and 2 triangles, etc.). There are exactly 17 ways to express 17 as the sum of 1 or more primes - 17 is the only integer which is equal to the number of prime partitions of itself. 17 is the number of wallpaper groups. 17 is the only prime of the form p^q + q^p, where p and q are prime. 17 is the only multidigit number n such that n + SOD(n) and n - SOD(n) are square numbers, where SOD means sum of digits. 17 is the number of syllables in a haiku. 17 is the smallest odd prime such that no odd Fibonacci number is divisible by it. 17 is the minimum number of Sudoku clues that produces a unique answer.

Today is the 168th day of the year.  Properties of 168 include:
 168 is a 57-gonal, abundant, amenable, arithmetic, composite, even, odious, pernicious, polite, practical, wasteful and Zumkeller number. 168 is the number of hours in a week. 168 is the number of primes less than 1000. 168 is the total number of spots on a set of dominoes. 168 can be expressed as the sum of three squares: 168 = 2^2+8^2+10^2.  168 can be expressed as the difference of two squares in three different ways: 168 = 17^2 – 11^2 = 23^2 – 19^2 = 43^2 – 41^2 It has 16 divisors, whose sum is σ = 480. Its totient is φ = 48.  The sum of its prime factors is 16 (or 12 counting only the distinct ones).  The product of its digits is 48, while the sum is 15. It is the short leg of a Pythagorean triple: 168^2 + 3526^2 = 3530^2. 168 is nontrivially palindromic and repdigit number when written in base 13, base 20, base 24, and base 27. It is a nude number because it is divisible by every one of its digits. 168 is an idoneal number. It is one of the 548 Lynch-Bell numbers. It is a plaindrome in base 5, base 10 and base 13.  It is a nialpdrome in base 4, base 6, base 7, base 13, base 14, base 15 and base 16.  It is a zygodrome in base 5 and base 13. It is a pseudoperfect number, because it is the sum of a subset of its proper divisors.

Today is the 1629th day of the decade.  The properties of 1629 include:
 1629 is a 47-gonal, 544-gonal, amenable, apocalyptic, composite, congruent, deficient, Curzon, Duffian, odd, odious, pernicious, polite, and wasteful number. It has 6 divisors, whose sum is σ = 2366. Its totient is φ = 1080.  The sum of its prime factors is 187 (or 184 counting only the distinct ones).  The product of its digits is 108, while the sum is 18. It can be written as a sum of positive squares in only one way, i.e., 1629 = 900 + 729 = 30^2 + 27^2.  It can also be written as the difference of two squares in three different ways: 1629 = 95^2 – 86^2 = 273^2 – 270^2 = 815^2 – 814^2. It is the short leg of a Pythagorean triple: 1629^2 + 147420^2 = 147429^2.  It is a nialpdrome in base 9 and base 13.

Today is the 5281st day of the century.  The properties of 5281 include:

 5281 is a prime number.  It is a weak prime, and an a-pointer prime.  It is the larger of a pair of twin primes. It is an amenable, arithmetic, centered decagonal, concentric 22-gonal, cyclic, deficient, happy, odd, odious, pernicious, polite and square-free number. It has 2 divisors, whose sum is σ = 5282. Its totient is φ = 5280.  The product of its digits is 80, while the sum is 16. It can be written as a sum of positive squares in only one way: 5881 = 3600 + 1681 = 60^2 + 41^2 .  It can also be written as the difference of two squares: 5281 = 2641^2 – 2640^2. It is short leg of a Pythagorean triple: 5281^2 + 13944480^2 = 13944481^2.  It is also the hypotenuse of a different Pythagorean triple: 5281^2 = 1919^2+4920^2.