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Extra! Extra! Hear All About It!

I am sure you have all read about it...In fact, I am sure it was in the headlines of your local newspapers or the lead story on local newscasts this month. Just to be sure, I am talking about the recently announced mathematics discovery by Christian Mauduit and Joël Rivat, two mathematicians from the Institut de Mathématiques de Luminy. In fact, I know all of you shared this discovery and its importance with your students...as an example that new mathematics continues to be created, etc.

To illustrate, consider this set of numbers: 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, and 2089. These are all the primes between 2000 and 2100. Now, find their digital sums (e.g. 2+0+0+3=5, 4, 10, 11, 13, 14, 12, 11, 17, 11, 13, 19, and 20. That is, for this small subset of prime numbers, five of the digital sums are even and nine of them are odd.

So what? Verifying a conjecture posed by Russian mathematician Alexandre Gelfond in 1968, Mauduit and Rivat proved that on average, the chance of a digital sum of a prime numbers being even equals the chance it is odd. In their proof, they used combinatorial mathematics, analytical number theory, and harmonic analysis. Furthermore, the claim is that their methods "are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers."

But, you ask, is there any real-world use of this new discovery in mathematics? As might be expected, the discovery connects to the important process of producing sequences of pseudo-random numbers and thus, the discovery will be applied in the areas of digital simulation and cryptography.

As a final thought, the value must lie in the proof techniques, not the announced discovery itself. For example, because the number of prime numbers is countably infinite, even if the number of even digital sums of prime numbers was one-billionth the number of odd digital sums of prime numbers, both sets would on average be equal since both would be countably infinite as well. Or, am I way off base?

Source: ScienceDaily, May 13, 2010