Numbers: Their Tales, Types, and Treasures.
Chapter 8: Special Numbers
There are a number of conjectures about prime numbers that have evolved over the years. Some conjectures have been proved, and some still remain open, such as the Goldbach conjecture and the twin primes conjecture mentioned earlier. Here are some other “facts” about prime numbers that have not been proved or disproved yet:
· There are infinitely many prime numbers of the form n2 + 1, where n is a natural number.
· There is always a prime number between n2 and (n + 1)2.
· There is always a prime number between n and 2n.
· There is an arithmetic progression of consecutive prime numbers for any given finite length, such as 251, 257, 263, 269, which has a length of 4. So far, the largest such length is 10.
· If n is a prime number, then 2n – 1 is not divisible by the square of a prime number.
· There are infinitely many prime numbers of the form n! – 1.
· There are infinitely many prime numbers of the form 2n – 1 (i.e., Mersenne primes).
· Every Fermat number 22n – 1 is a composite number for n > 4.
· The Fibonacci numbers (see chapter 6, section 1) contain an infinite number of prime numbers. Here are some of these: 2; 3; 5; 13; 89; 233; 1,597; 28,657; 514,229; 433,494,437; 2,971,215,073; and 99,194,853,094,755,497.
The study of primes is boundless; we have merely shown some of the peculiarities that can be discovered among the prime numbers. Yet there are many other peculiarities that readers may want to discover on their own.