About a century ago, mathematicians proved that there are infinitely many prime numbers among numbers of the form a2 + k*b2, where a and b are integers and k is a certain constant. But only recently have they been able to prove that there are infinitely many primes among numbers of the form a2 + b4. The proof was given by John Friedlander of the University of Toronto and Henryk Iwaniec of Rutgers University. They also determined the likely number of such primes within a given range. Among whole numbers up to 100, there are only 6 such prime numbers: 2, 5, 17, 37, 41, and 97.
Here are a couple of brain teasers:
Each interval between consecutive squares contains at least one prime. Can you prove it or give a counter-example? If so, e-mail me and I will be very amazed.
The number of twin primes, i.e. pairs of primes that differ by 2 (such as 17 and 19), is infinite. Can you prove it or disprove it? In either case, publish!
A banker and mathematician, Andrew Beal, is offering a $50,000 prize for a proof of his conjecture or $10,000 for a counterexample. The conjecture, similar to the equation given in Fermat's Last Theorem, is as follows:
Given the equation Ax + By = Cz, where A, B, C, x, y, and z are whole numbers and x, y, and z are greater than 2, there exist only solutions such that A, B, and C have a common factor.
For example: 36 + 183 = 38, where 3, 18, and 3 all have the common factor 3.
From a letter by Prof. Alar Toomre, an applied mathematician and theoretical astronomer, to a colleague at MIT:
"Our discussion about Hardy and Ramanujan's
1729 = 123 + 13 = 103 + 93
prompted me today to use my home computer to examine what I had long intended but had never gotten around to:
Granted that 1729 is their smallest specimen, how common or rare are the integers expressible as sums of simple cubes in two different ways?
Well, it turns out that the first million integers contain 25 such examples, not including obvious "duplicates" like 13,832 = 8 x 1729, nor any others 8 or 27 or 64 or 125 . . . times those already known. Even including such pale imitations, the grand total comes only to 43.
The first example beyond 1729 is
4104 = 163 + 23 = 153 + 93.
This number seems to me the prettiest of them all, since it is plainly divisible BOTH by 8 and by 27. One of its remnant factors then equals 513 = 83 + 13, whereas the other yields 152 = 53 + 33 just as sweetly.
No such kind words apply to the next non-trivial example
20,683 = 273 + 103 = 243 + 193
nor to the last one
994,688 = 993 + 293 = 923 + 603.
Now you find us the first integer expressible as a sum of cubes in three different ways! Actually I have a marvelous proof that none exist, but alas the margin of this page is far too narrow to contain it."
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