Pretty Numbers Moloy De !

6 6 is the smallest perfect number 1, 2 and 3 are the only proper positive divisors of 6 and = 6 6, 28, 496, and 8128 are few other perfect numbers 2 p−1 (2 p − 1) gives an even perfect number whenever 2 p − 1 is prime (Mersenne Prime) It is unknown whether there are any odd perfect numbers or not

7 7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass A regular n-gon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and any number of distinct Fermat Primes

8 8 is the largest cube in the Fibonacci sequence Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34,.. The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences, long before it was known in Europe. Developments have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopāla (c.1135 AD), and Hemachandra (c.1150 AD)

16 16 is the only number of the form X power Y = Y power X, where X and Y are different integers 16 = 2 power 4 = 4 power 2

25 25 is the smallest square that can be written as a sum of 2 squares 25 = (9, 16, 25) is called a Pythagorean Triplet Smallest Pythagorean Triplet using same digits: =

26 26 is the only positive number to be directly between a square and a cube = 26 = Catalan's Conjecture or Mihăilescu's Theorem: 8 (2 cube) and 9 (3 square) are the only consecutive powers of natural numbers

27 27 is the largest number that is the sum of the digits of its cube 27 cube = = 27 Below is the complete list: 0 cube = 0 1 cube = 1 8 cube = cube = cube = cube = cube = 19683

48 48 is the smallest number with 10 divisors 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 are the only divisors of 48

54 54 is the smallest number that can be written as the sum of three squares in three ways 54 = = = Following are few more such examples: 54 (3 ways) 129 (4 ways) 194 (5 ways) 209 (6 ways) 341 (7 ways) 374 (8 ways) 614 (9 ways) 594 (10 ways) 854 (11 ways)

65 65 is the smallest number that becomes square if its reverse is either added to or subtracted from it = 121 = = 9 = 3 2 Next such example is = = – = = 738 2

96 96 is the smallest number that can be written as the difference of two squares in four ways = = = = = = = = 96 Following are few more such examples: 96 (4 Ways), 192 (5 Ways), 240 (6 Ways), 576 (7 Ways), 480 (8 Ways), 720 (9 Ways), 960 (10 Ways),..

is the smallest square which is also the sum of 4 consecutive cubes 10 2 = 100 = Next such square number (if it exists) is bigger than 1,00,000 However, 6 2 = 36 = = =

is the largest known number n with the property that n - 2 power k is prime for all possible positive integral values of k for which n - 2 power k is positive k = 1: = 103 is a prime. k = 2: = 101 is a prime. k = 3: = 97 is a prime. k = 4: = 89 is a prime. k = 5: = 73 is a prime. k = 6: = 41 is a prime. 4, 7, 15, 21, 45, 75 and 105 are the only such numbers known till today

is a value of n for which n! + 1 is prime The value of 116! is of the order of 10 power 190 n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, No other factorial prime is known till today

is the only known square number of the form 1 + p + p 2 + p 3 + p 4 where p is prime 11 2 = 121 = = = 400 =

is the only known number that contains all its proper divisors as its proper substrings 5 and 25 are the only proper divisors of is the only number where the count of alphabets in its name “FOUR” equals the number 4

is a Mersenne Prime named after 17th century French scholar Marin Mersenne 127 = (2 power 7) – 1, is a prime number and is of the form M(n) = (2 power n) - 1 As of October 2009, only 47 Mersenne Primes are known. The largest known prime number M( ) is a Mersenne Prime with more than 10 million digits M(2) = 3 M(3) = 7 M(5) = 31 M(7) = 127 M(13) = 8191 M(17) = M(19) =

is the largest number that cannot be expressed as a sum of distinct squares There are only 31 numbers that cannot be expressed as the sum of distinct squares: 2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112 and 128 Following are few examples of other type: 129 = = = = = =

is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits = =

is the sum of the cubes of the digits of the sum of the cubes of its digits 1 cube + 3 cube + 6 cube = = cube + 4 cube + 4 cube = = 136 A different type: 3435 = = 3 power power power power 5

is a Factorion A Factorion is equal to the sum of factorials (n! = 1*2*3*...*n) of its digits There are exactly four Factorions: 1 = 1! 2 = 2! 145 = = 1! + 4! + 5! = = 4! + 0! + 5! + 8! + 5!

is a palindromic prime Prime: Not divisible by any integer except 1 and itself Palindrome: Reads the same backward and forward The first few palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, … Please note that there is no 4-digit palindromic prime.

is the sum of all the primes between its smallest and largest prime factors 155 = 5 times = 155 Following are few more of such examples: 10 = 2 times 5 39 = 3 times = 5 times = 7 times 53

157 Square of 157 contains the same digits as that in the square of square = square = Following are few more of such examples: 13 square = square = square = square = square = square = square = square = square = square = square = square = square = square = square = square = square = square =

is the smallest number that can be written as the sum of 4 positive squares in 9 ways Following are few more of such examples: 178 (9 ways) 198 (10 ways) 202 (11 ways) = = = = = = = = = 162

is the midpoint of the n-th larger prime and n-th smaller prime, for 1 ≤ n ≤ 6 ( ) / 2 = 165 ( ) / 2 = 165 ( ) / 2 = 165 ( ) / 2 = 165 ( ) / 2 = 165 ( ) / 2 = 165

is the smallest number whose 4th power ( ) begins with 4 identical digits

square is and it contains only 2 distinct digits

has the cube ( = 178 cube) with the same digits as another cube ( = 196 cube)

is the smallest number n so that n (183) concatenated with n+1 (184) is a square ( = 428 square)

is the smallest number that is not known to reach a palindrome when repeatedly added to its reverse

= = = = =