1. Pretty Numbers Moloy De !

2. 6 6 is the smallest perfect number 1, 2 and 3 are the only proper positive divisors of 6 and 1 + 2 + 3 = 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

3. 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

4. 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)

5. 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

6. 25 25 is the smallest square that can be written as a sum of 2 squares 25 = 9 + 16 (9, 16, 25) is called a Pythagorean Triplet Smallest Pythagorean Triplet using same digits: 1890 2 + 19800 2 = 19890 2

7. 26 26 is the only positive number to be directly between a square and a cube 25 + 1 = 26 = 27 - 1 Catalan's Conjecture or Mihăilescu's Theorem: 8 (2 cube) and 9 (3 square) are the only consecutive powers of natural numbers

8. 27 27 is the largest number that is the sum of the digits of its cube 27 cube = 19683 1 + 9 + 6 + 8 + 3 = 27 Below is the complete list: 0 cube = 0 1 cube = 1 8 cube = 512 17 cube = 4913 18 cube = 5823 26 cube = 17576 27 cube = 19683

9. 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

10. 54 54 is the smallest number that can be written as the sum of three squares in three ways 54 = 1 + 4 + 49 54 = 9 + 9 + 36 54 = 4 + 25 + 25 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)

11. 65 65 is the smallest number that becomes square if its reverse is either added to or subtracted from it 65 + 56 = 121 = 11 2 65 - 56 = 9 = 3 2 Next such example is 621770 621770 + 77126 = 698896 = 836 2 621770 – 77126 = 544644 = 738 2

12. 96 96 is the smallest number that can be written as the difference of two squares in four ways 10 2 - 2 2 = 100 - 4 = 96 11 2 - 5 2 = 121 - 25 = 96 14 2 - 10 2 = 196 - 100 = 96 25 2 - 23 2 = 625 - 529 = 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),..

13. 100 100 is the smallest square which is also the sum of 4 consecutive cubes 10 2 = 100 = 1 3 + 2 3 + 3 3 + 4 3 Next such square number (if it exists) is bigger than 1,00,000 However, 6 2 = 36 = 1 3 + 2 3 + 3 3 204 2 = 41616 = 23 3 + 24 3 + 25 3

14. 105 105 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: 105 - 2 = 103 is a prime. k = 2: 105 - 4 = 101 is a prime. k = 3: 105 - 8 = 97 is a prime. k = 4: 105 - 16 = 89 is a prime. k = 5: 105 - 32 = 73 is a prime. k = 6: 105 - 64 = 41 is a prime. 4, 7, 15, 21, 45, 75 and 105 are the only such numbers known till today

15. 116 116 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, 26951. 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, 34790. No other factorial prime is known till today

16. 121 121 is the only known square number of the form 1 + p + p 2 + p 3 + p 4 where p is prime 11 2 = 121 = 1 + 3 + 9 + 27 + 81 = 1 + 3 + 3 2 + 3 3 + 3 4 20 2 = 400 = 1 + 7 + 7 2 + 7 3

17. 125 125 is the only known number that contains all its proper divisors as its proper substrings 5 and 25 are the only proper divisors of 125 4 is the only number where the count of alphabets in its name “FOUR” equals the number 4

18. 127 127 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(43112609) 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) =131071 M(19) = 524287

19. 128 128 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 = 100 + 25 + 4 130 = 121 + 9 131 = 121 + 9 + 1 132 = 81 + 25 + 16 + 9 + 1 133 = 81 + 36 + 16 134 = 121 + 9 + 4

20. 132 132 is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits 13 + 12 + 31 + 32 + 21 + 23 = 132 35964 = 359 + 356 + 354 + 395 + 396 + 394 + 365 + 369 + 364 + 345 + 349 + 346 + 539 + 536 + 534 + 593 + 596 + 594 + 563 + 569 + 564 + 543 + 549 + 546 + 935 + 936 + 934 + 953 + 956 + 954 + 963 + 965 + 964 + 943 + 945 + 946 + 635 + 639 + 634 + 653 + 659 + 654 + 693 + 695 + 694 + 643 + 645 + 649 + 435 + 439 + 436 + 453 + 459 + 456 + 493 + 495 + 496 + 463 + 465 + 469

21. 136 136 is the sum of the cubes of the digits of the sum of the cubes of its digits 1 cube + 3 cube + 6 cube = 1 + 27 + 216 = 244 2 cube + 4 cube + 4 cube = 8 + 64 + 64 = 136 A different type: 3435 = 27 + 256 + 27 + 3125 = 3 power 3 + 4 power 4 + 3 power 3 + 5 power 5

22. 145 145 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 + 24 + 120 = 1! + 4! + 5! 40585 = 24 + 1 + 120 + 40320 + 120 = 4! + 0! + 5! + 8! + 5!

23. 151 151 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.

24. 155 155 is the sum of all the primes between its smallest and largest prime factors 155 = 5 times 31 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 = 155 Following are few more of such examples: 10 = 2 times 5 39 = 3 times 13 155 = 5 times 31 371 = 7 times 53

25. 157 Square of 157 contains the same digits as that in the square of 158 157 square = 24649 158 square = 24964 Following are few more of such examples: 13 square = 169 14 square = 196 436 square = 190096 437 square = 190969 515 square = 265225 516 square = 266256 847 square = 717409 848 square = 719104 863 square = 744769 864 square = 746496 900 square = 810000 901 square = 811801 913 square = 833569 914 square = 835396 987 square = 974169 988 square = 976144 992 square = 984064 993 square = 986049

26. 162 162 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) 1. 1 + 1 + 16 + 144 = 162 2. 1 + 4 + 36 + 121 = 162 3. 1 + 16 + 64 + 81 = 162 4. 1 + 25 + 36 + 100 = 162 5. 4 + 9 + 49 + 100 = 162 6. 9 + 16 + 16 + 121 = 162 7. 9 + 25 + 64 + 64 = 162 8. 9 + 36 + 36 + 81 = 162 9. 16 + 16 + 49 + 81 = 162

27. 165 165 is the midpoint of the n-th larger prime and n-th smaller prime, for 1 ≤ n ≤ 6 (137 + 193) / 2 = 165 (139 + 191) / 2 = 165 (149 + 181) / 2 = 165 (151 + 179) / 2 = 165 (157 + 173) / 2 = 165 (163 + 167) / 2 = 165

28. 167 167 is the smallest number whose 4th power (777796321) begins with 4 identical digits

29. 173 173 square is 29929 and it contains only 2 distinct digits

30. 178 178 has the cube (5639752 = 178 cube) with the same digits as another cube (7529536 = 196 cube)

31. 183 183 is the smallest number n so that n (183) concatenated with n+1 (184) is a square (183184 = 428 square)

32. 196 196 is the smallest number that is not known to reach a palindrome when repeatedly added to its reverse

33. 198 198 = 11 + 99 + 88 198 = 19 + 98 + 81 = 91 + 89 + 18 2997 = 222 + 999 + 999 + 777 13332 = 1111 + 3333 + 3333 + 3333 + 2222