What Do Mathematicians Think About?
The answer: A Lot! Let's look at two examples that can be shared with students. First, "twin primes" are two consecutive prime numbers that differ by 2. Some examples are 11 and 13, 29 and 31, or 41 and 43. The largest known twin primes are (2003663613)(2^{195000}) ± 1, each with 58711 decimal digits.
Second, a number p is a "cluster prime" if every even number less than p2 is the difference of two primes (both less than or equal to p). For example, 13 is a cluster prime because 10=133, 8=113, 6=115, 4=117, and 2=1311. The smallest noncluster prime is 97.
The questions mathematicians are thinking about: Are there infinitely many twin primes? Are there infinitely many cluster primes? Erdos et al conjecture that there are more twin primes than cluster primes. All three questions are considered to be very elusive problems in number theory.
If you want to read more about these questions, a good source is the Wolfram MathWorld. Check out twin prime and cluster prime.
Source: Adapted from Science News, February 6, 1999, p. 95
