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.

Primes in arithmetic progression, such one is a permutation of the other

Look at 1487, 4817 and 8147. 
They are three primes with the same digits, one is a permutation of the other, and are in arithmetic progression with a common difference of 3300.  

Another examples:

Common difference, first term, second term and last term

3330    1487     4817     8147
3330     2969    6299     9629
3330     11483    14813    18143
30222    11497    41719    71941
504       12713    13217    13721
4500    12739    17239    21739
4500    12757    17257    21757
4500    12799    17299    21799
33300    14821    48121    81421
16650    14831    31481    48131
32292    14897    47189    79481
33300    18503    51803    85103
33300    18593    51893    85193
15948    19543    35491    51439
450    20161    20611    21061
4950    20353    25303    30253    35203*
4950    20359    25309    30259    35209*
3330    20747    24077    27407
4500    23887    28387    32887
27720    25087    52807    80527
33480    25793    59273    92753
13608    25913    39521    53129
33300    25981    59281    92581
4950    26317    31267    36217
33030    26597    59627    92657
450    28933    29383    29833
33300    29669    62969    96269
3330    31489    34819    38149
8352    31489    39841    48193
30330    32969    63299    93629
4500    34961    39461    43961
4950    35407    40357    45307
4050    35491    39541    43591
17946    35671    53617    71563
14076    37561    51637    65713
4950    49547    54497    59447
450    55603    56053    56503
3330    60373    63703    67033    70363*
4950    60757    65707    70657    75607*
3330    61487    64817    68147
3330    62597    65927    69257
4950    62773    67723    72673    77623*
450    63499    63949    64399
450    67829    68279    68729
9450    68713    78163    87613
2772    71947    74719    77491
5004    73589    78593    83597
450    76717    77167    77617
4950    76819    81769    86719
5238    78941    84179    89417
8910    80191    89101    98011
4950    83987    88937    93887    98837 (all primes)
4950    88937    93887    98837
4500    89387    93887    98387
450    92381    92831    93281

*the last term is not prime