## lunes, 25 de octubre de 2010

### My contributions to Prime Curios

Here are my contributions to Prime curios

11  : The smallest prime which when sandwiched between a two-digit repdigit gives a multiple of 11. In other words 1111, 2112, 3113, 4114, 5115, 6116, 7117, 8118, and 9119 are multiples of 11.

23   : 23 = -22 + 33

43   : 43 =  42 + 33

97   : 97 and its double (194) and triple (291) use the same number of characters (five) when expressed in Roman numerals: XCVII, CXCIV, and CCXCI.

109  : The smallest non-trivial prime that is the sum of the reversal of two consecutive primes (109 = R(47) + R(53) = 74 + 35).

239  : 1+3+5+7+....+237+239 = 239+241+243+...+335+337. Note that 239 and 337 are both primes.

251 : The 251st Fibonacci number (F251) has a sum of digits equal to 251. The two smaller prime numbers with this property are 5 and 31

269  : The 269th day of a non-leap year is 26 September (26/9)

617  : 617 = 1!2 + 2!2 + 3!2 + 4!2

991  : 9912 = 982081 and 982 + 0 + 8 + 1 = 991.

1009 : The sum of digits of 1009 is a substring of itself and of its square.

1201  12012 = 601+602+603...+1799+1800+1801. With 1201, 601, and 1801 each being prime

1669 : 16692 = 2785561, and 278 * (5/5) * 6 + 1 = 1669

1669 : The smallest prime  that appears in the same position of its own value when the Roman numerals  (from 1 to 3999) are placed in lexicographic order. The other primes with this property are 3623 and 3631

4027  40275 = 33015 + 31695 + 30375 + 24115 + 14815 + 8595 + 5695. Note that all base numbers and exponents are prime. Found by Takao Nakamura.

4561  : The digits of 4561 (abcd) produce a distinct nine-digit product in the following expression: (a+b+c+d)(ab+cd)(a+bcd)(abc+d)

6833 : 68332 = 46689889, and 4 * 6 + 6898 - 89 = 6833.

8209 : 82093 = 553185473329, and 52 + 52 + 32 + 12 + 852 + 42 + 72 + 32 + 32 + 292 = 8209.

12637 : The smallest prime such that the differences between the 5 consecutive primes starting with it   are (4,6,6,6): 12637, 12641, 12647, 12653, 12659.

15017 : 15017 = 1!2+2!2+3!2+4!2+5!2

17783 : The smallest prime which is the sum of two, three, four, and five consecutive composite  numbers:
17783 = 8891 + 8892 = 5926 + 5928 + 5929 = 4444 + 4445 + 4446 + 4448 =
3554 + 3555 + 3556 + 3558 + 3560.

28567 : is the smallest prime, which is a Fibonacci number (F(23)prime) and an anagram of a triangular number (67528 = T(367)prime).

41579 : is the only prime p, such that p and p expressed in some base < 10, taken together are   pandigital. 41579 = 63028 in base 9.

38981039 : The smallest number whose square begins and ends with the same seven digits: 389810392 = 1519521401519521.

989450477 : The log730 (989450477) starts out equal to the first dozen digits of pi.

298999999999 : The smallest prime with sum of digits equal to 100.