Facts about 2017

10 interesting facts about 2017

1. \(2017\) is a prime \(\big(\)the \(306^\text{th}\big).\)

2. \( 2017 ^ 2 - 2017 + 1 \) is also a prime.

3. \( 2017 = 1009^2 - 1008^2 \) is the only way to express it as the difference of 2 squares.

4. \(2017 = 44^2 + 9^2 = 37^2 + 2 \times 18^2 = 17^2 + 3 \times 24^2 = 9^2 + 4 \times 22^2\). It cannot be written in the form \( n^2 + 5 \times m^2 \).

5. \( 2017 = 11^3 + 7^3 + 7^3 = 15 ^3 - 3^3 - 11^3 \).

6. With 63 straight cuts on a circular pizza, each cut going through 2 points on the circumference, we get a maximum of 2017 pieces.

7. \( \sqrt[3]{2017} = 12.63480759\ldots \), and 2017 is the first positive integer the first 10 digits of whose cube root are the distinct digits 0 to 9.

8. \( 2017 = \frac{ 63 \times 64 } { 2} + 1 \) is a central polygonal number.

9. There are 2017 ways to place the integers from 1 to 16 in a \( 4 \times 4 \) grid so that each row, column, and both diagonals are increasing.

10. 2 and 3 are quadratic residues mod 2017: \(986^2 \equiv 2 \pmod{2017}\) and \(258^2 \equiv 3 \pmod {2017}.\)

Additional facts about 2017

1. \( \phi(2017) = \phi (2016) + \phi (2015) \), so 2017 is a Fibonacci number.

2. \( 2017, \frac{ 2017 + 1 } { 2} , \frac{ 2017 + 2 }{ 3} \) are all primes.

3. \( 2017 \) in bases 31 and 32 are each a palindrome: \( 2017_{10} = 232_{31} = 1v1_{32} \).

4. \(2017= 29^2 + 6 \times 14^2 = 15^2 + 7 \times 16^2 = 37^2 + 8 \times 9^2 = 44^2 + 9 \times 3^2\).

5. \( 2017 = 2^{11} - (11^{\text{th}} \text{ prime}) \).

6. \( 2017 =2^{11} - 2^5 + 1 \) is a Solinas prime.

7. \(2017\pi\) rounded to the nearest integer is a prime.

8. \(2017e\) rounded to the nearest integer is a prime.

9. The sum of all primes till 2017 is a prime number.

10. The sum of the cubes of the differences between adjacent primes till 2017 is prime: i.e. \((3-2)^3 + (5-3)^3 + (7-5)^3 + \cdots\) is prime.

11. The prime before 2017 is \(2017+(2-0-1-7)\) making it a sexy prime, and the prime after 2017 is \(2017+(2+0+1+7)\).

12. Inserting 7 between any two adjacent digits in 2017 results in a prime number: i.e. 20177, 20717, 27017 are all primes.

13. \(2017_8\) is a prime number.

14. \(2017 = 12^3 + 6^3 + 4^3 + 2^3 + 1^3\).

15. 20170123456789 is a prime number.

16. The 2017\(^\text{th}\) prime is 17359, and 201717359 is a prime.

17. \(2017 = 44^2 + 3^4\).

18. \(\phi (2017) = 2 \times \phi(2018)\). This is equivalent to the fact that both \(2017\) and \(\frac{2017+1}{2}\) are prime.

19. \(2017 = 11111100001_2 = 100000000000_2 - 100000_2 +1.\)

20. \(2017 = 19\cdot 22 + 22\cdot 39 + 39\cdot 19\). In a form of \(2017 = xy+yz+zx\), the minimum sum of positive integers \(x+y+z = 19+22+39=80\).