## Tuesday, 17 January 2012

### Tuesday, 17 January

Today is 17 day of the year.
17 is prime.
17 = 2 + 3 + 5 + 7 i.e the sum of the first four primes.
17 = 2^3 + 3^2. 17 is the only prime of the form p^q + q^p where p and q are both prime.
17 is the most random number.
17 is a member of one primitive, Pythagorean triple (8, 15, 17).
17 is the third Fermat number defined by the form 2^2^n + 1.
17 is the seventh Tribonacci number.

17 is the only prime that can be expressed as the sum of four consecutive primes

Proof
17 is the sum of the first four consecutive primes.
The first four consecutive primes are the only set of four consecutive primes that include the number 2.
2 is the only even prime.
Therefore, all other sets of four consecutive primes have the characteristic that they are all odd numbers.
All odd numbers are of the form 2a + 1.
Assume we have four primes that are 2a + 1, 2b + 1, 2c + 1 and 2d + 1.
Their sum s = 2a + 1 + 2b + 1 + 2c + 1 + 2d + 1
s = 2(a + b + c + d) + 4 = 2(a + b + c + d + 2)
Thus the sum of any four consecutive primes, apart from the first four, is even.
If the sum is even then it cannot be prime.
QED