The word Fraction comes from the Latin word fractio which means ‘to break’.

The Riemann Hypothesis

The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the “Riemann zeta function” all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.

The Twin Prime conjecture

The twin primes conjecture concerns pairs of prime numbers with a difference of 2. The numbers 5 and 7 are twin primes. So are 17 and 19. The conjecture predicts that there are infinitely many such pairs among the counting numbers, or integers. Mathematicians made a burst of progress on the problem in the last decade, but they remain far from solving it.

Yang-Mills theory and the quantum mass gap

One of the major underpinnings of modern quantum mechanics is Yang-Mills theory, which describes the quantum behavior of electromagnetism and the weak and strong nuclear forces in terms of mathematical structures that arise in studying geometric symmetries. The predictions of Yang-Mills theory have been verified by countless experiments, and the theory is an important part of our understanding of how atoms are put together. Despite that physical success, the theoretical mathematical underpinnings of the theory remain unclear. One particular problem of interest is the “mass gap,” which requires that certain subatomic particles that are in some ways analogous to massless photons instead actually have a positive mass. The mass gap is an important part of why nuclear forces are extremely strong relative to electromagnetism and gravity, but have extremely short ranges.

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