# 2017

### Primes

• 2017 is a prime number (no positive integer divides it except 1 and itself).
• Half and third of 2017 is also a prime after rounding up: 1009 and 673 are primes. Increasing 2017 by integer parts of its third or two thirds gives 2689 and 3361 which are also primes.
• 2017 remains prime when the digit 1 or 7 is inserted between any two digits, that is 21017, 20117, 27017, 20717, 20177 are all primes.
• 2017 remains prime when the digit 0 is inserted into the three places between the digits, that is 2000107 is prime.
• Sum of the odd primes not greater than 2017 is 283079, which is prime.
• Sum of the cube of the gaps between the consecutive primes not greater than 2017 is 258569, which is prime.
• The next integer is even, but double of a prime (2018 = 2 × 1009) and the previous integer is divisible by every nonzero digits of 2017.
• 2017 is one of the primes that the next prime can be obtained from it by adding the sum of the digits to it (2027 = 2017 + 10). 2017 is even the smallest prime that the sequence of consecutive six primes contains three pairs of primes with difference 10: 2017, 2027, 2029, 2039, 2053, 2063.
• 2017 is the difference between the 11-th 2-power and the 11-th prime: 2017 = 211 - 31.
• 2017 considered in base 8 equals 1039, which is prime.
• Multiplying 2017 by the two most famous math constants (π ≈ 3.14159, e ≈ 2.718281828) and rounding to the closest integer we get primes (round(2017π) = 6337, round(2017e) = 5483).
• Concatenating 2017 with the 2017-th prime gives 201717539, which is prime.
• The digits of the binary form of 2017 in reverse order is prime (2017 = 111111000012, 100001111112 = 1087) and 1 plus the riversal of 2017 is prime (7103).

### Counts 2017

• There are 2017 ways of writing the numbers 1, 2,…, 16 into a 4 × 4 table such that each row, each column and both diagonals are increasing.
• The order of the 7 dwarfs is Bashful, Doc, Dopey, Grumpy, Happy, Sleepy, Sneezy. They can be stand in a row without keeping the original order of any three consecutive dwarfs in 2017 ways. E.g. Doc, Sneezy, Dopey, Happy, Sleepy, Bashful, Grumpy is not counted, because Doc, Dopey, Grumpy are in the original order.
• 32 angular sectors divide the planes into at most 2017 regions.
• The number of the 27-digit 7-th powers is 2017.

### Other beauties of 2017

• 2017 is an element of a Pythagorean triple, namely 20172 = 7922 + 18552, that is there is a right triangle with integer sides and hypotenuse length 2017.
• The first 10 digits of the cube root of 2017 (12.63480759) contain all digits from 0 to 9. 2017 is the smallest positive integer with this property.
• The nicest power-sums: 2017 = 92 + 442, 2017 = 73 + 73 + 113, 2017 = 63 + 73 + 93 + 93, and a few more: 2017 = 442 + 34 = 172 + 123, 2017 = 122 + 282 + 332 = 212 + 262 + 302 = 182 + 182 + 372, 2017 = 13 + 23 + 23 + 103 + 103 = 13 + 23 + 43 + 63 + 123 = 23 + 43 + 63 + 93 + 103.

More interesting facts can be found on The On-line Encyclopedia of Integer Sequences.