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.