## 2017 is not just another prime number

Posted by tjwei on 星期日, 1月 01, 2017 with No comments
Good bye year 2016. Hello year 2017.
We all know that 2017 is a prime number, but it is more than just another prime number.

• 2017π (rounds to nearest integer) is a prime
• 2017e (rounds to nearest integer ) is a prime.
• The sum of all odd primes up to 2017 is a prime number, i.e. 3+5+7+11+...+2017 is a prime number.
• The sum of the cube of gap of primes up to 2017 is a prime number. That is (3-2)^3 + (5-3)^3 + (7-5)^3 + (11-7)^3 + ... + (2017-2011)^3 is a prime number.
• The prime number before 2017 is 2017+(2-0-1-7), which makes it a sexy prime, and the prime after 2017 is 2017+(2+0+1+7). 2017 itself is of course equal to  2017+(2*0*1*7)
• Insert 7 into any two digits of 2017, it is still a prime number, i.e. 27017, 20717, 20177 are all primes. Plus, 20177 is also a prime number
• Since all digits of 2017 is less than 8, it can be viewed as an octal. 2017 is still a prime number as an octal.
• 2017 can be written as a sum of three cubes of primes, i,e,  p^3 +q^3 +r^3 for some primes p, q, r.
• 2017 can be written as a sum of cubes of five distinct integers.
• 2017 can be written as  x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+4y^2 x^2+6y^2, x^2+7y^2, x^2+8y^2, x^2+9y^2 (for positive integers x, y)
• 20170123456789 is also a prime
• the 2017th prime number is 17539 and 201717539 is also a prime.
• Let p=2017, then both (p+1)/2 and (p+2)/3 are prime numbers.
• The first ten digits of the decimal expansion of the cubic root of 2017 contains all different digits 0~9. 2017 is the least integer has this property.
• 2017 = 2^11 - 11th prime

You can check OEIS for more interesting facts for your favorite numbers :)

Update: a sagemath worksheet to verify these facts by William Stein
Categories: