Numbers: Their Tales, Types, and Treasures.

Chapter 8: Special Numbers

8.4.UNSOLVED QUESTIONS

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.