Árpád Kurusa mathematician, associate professor Department of Geometry Bolyai Institute Faculty of Science University of Szeged

## 2017

A bit late, but this note is about the number 2017 of this year.

Before the list comes observe that 2017 is prime number.

• $[2017\cdot \pi+0.5]$ is a prime;
• $[2017\cdot e+0.5]$ is a prime.
• The sum of all odd primes up to 2017 is a prime number
• The sum of the cube of gap of primes up to 2017 is a prime number. That is $(3-2)^3 + (5-3)^3 + (7-5)^3 + (11-7)^3 + \dots + (2017-2011)^3$ is a prime number;
• The prime number before 2017 is $2017+(2-0-1-7)$, and the prime after 2017 is $2017+(2+0+1+7)$;
• 27017, 20717, 20177, 20177 are all primes.
• 2017 is still a prime number if read as an octal;
• 2017 can be written as a sum of three cubes of primes;
• 2017 can be written as a sum of cubes of five distinct integers;
• 2017 can be written as $x^2+y^2$, $x^2+2y^2$, $x^2+3y^2$, $x^2+4y^2 x^2+6y^2$, $x^2+7y^2$, $x^2+8y^2$, $x^2+9y^2$, where $x,y$ are positive integers;
• 20170123456789 is also a prime
• the 2017th prime number is 17539 and 201717539 is also a prime;
• $(2017+1)/2$ and $(2017+2)/3$ are also prime numbers;
• 2017 is the least integer the first ten digits of the decimal expansion of the cubic root of which contains all different digits 0-9;
• $2^{11}-2017$ is just the $11$th prime.

To verify these facts one only has to see the SageMath worksheet by William Stein.