I normally don’t like The Economist as it promotes free trade between nations and historically that was the reason it started publishing and became popular. I however liked its description of the detection of gravitational waves:

TWO black holes circle one another. Both are about 100km across. One contains 36 times as much mass as the sun; the other, 29. They are locked in an orbital dance, a kilometre or so apart, that is accelerating rapidly to within a whisker of the speed of light. Their event horizons—the spheres defining their points-of-no-return—touch. There is a violent wobble as, for an instant, quintillions upon quintillions of kilograms redistribute themselves. Then there is calm. In under a second, a larger black hole has been born.

…

And then, 1.3 billion years later, in September 2015, on a small planet orbiting an unregarded yellow sun, at facilities known to the planet’s inhabitants as the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO), the faintest slice of those waves was caught. That slice, called GW150914 by LIGO’s masters and announced to the world on February 11th, is the first gravitational wave to be detected directly by human scientists. It is a triumph that has been a century in the making, opening a new window onto the universe and giving researchers a means to peer at hitherto inaccessible happenings, perhaps as far back in time as the Big Bang.

So a new largest prime number has been found.  It is

2^74,207,281 −  1

Of course we know that there is an infinitude of primes, so the number above is just the largest known.

A standard proof is via contradiction. Assume there is a largest prime p_n . Then it can be shown that the number p₁p₂p₃ … p_n + 1 is also a prime, contradicting our assumption that there is a largest prime.

Yesterday I found another proof from Wikipedia which is fascinating. There is a formula:

π/4 = 3/4 × 5/4  × 7/8 × 11/12 × 13/12 × 17/16 × 19/20 × 23/24 × 29/28 × 31/32 × …

The numerator is all primes (except 2), one after the other. The denominator is the nearest multiple of 4 of the numerator.

We know  π is irrational. If there are a finite number of primes, the right hand side is a rational number, which doesn’t makes sense since the left hand side is irrational. Hence the right hand side is an infinite product. Hence there is an infinitude of primes!

This month, a century ago, Albert Einstein wrote down the gravitational field equations. General Relativity is still the most beautiful theory discovered by the human intellect. There are two nice articles on this:

1. The New York Times: A Century Ago, Einstein’s Theory of Relativity Changed Everything.
2. The Economist: The most beautiful theory

The most fascinating result of General Relativity is that the universe is dynamic. Both articles give a brief history of what was on Einstein’s mind. When he found that his equations tell us that the universe is dynamic (i.e., either expanding or contracting), Einstein could not come to terms with it, initially. He then added a term to his field equations, perfectly allowed, to produce the result that the universe is static. This is called the cosmological constant term. However, Willem de Sitter came up with a solution for the field equations with this cosmological constant term, in which the universe is dynamic. This probably embarrassed Einstein, who later called it (adding the cosmological constant term) the greatest blunder of his life. However, physicists later concluded otherwise because there is no reason why this term shouldn’t be there. At any rate, Einstein’s General Relativity tells us that the universe is dynamic and only later did experiments verify this fact.

General Relativity also gives us a starting point – a scientific nature of inquiry – in asking about life, universe and everything.  In a film, A Brief History of Time, Roger Penrose says:

I think I would say that the universe has a purpose, it’s not somehow just there by chance … Some people, I think, take the view that the universe is just there and it runs along – it’s a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don’t think that’s a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it.