Goal: 400 words researched and written in half and hour. For me, for practice. Corrections welcome in the comments.

With Valentine’s Day upon us a seemingly appropriate mathematical topic is sexy primes. These numbers surely have some attraction to those in the mood for love.
In fact, the “sexy” in “sexy primes” is a reference to the Latin word for the number six: “sex”.
Sexy primes are prime numbers separated by six places from another prime number. You may remember that a prime number is a number that can only be divided by 1 and itself. So 5 is prime because you can only divide it by 1 (to give 5) or 5 (to give 1). All other numbers divide into 5 to give answers which are not whole numbers. Try it: 5 divided by 2 gives 2.5 – not a whole number.
In fact, 5 is not just prime, it’s sexy too. That’s because 11 is also a prime and is six places from 5.
The next sexy prime pair? 7 and 13. Then, 11 and 17. Actually, that’s the second time we’ve seen 11, isn’t it? It’s six places above 5 and six below 17, and all three are sexy primes. This means we’ve found our first sexy prime triplet: three primes with common difference six. The second of these is 7, 13 and 19. You can get sexy quadruplets too and one sexy quintuplet, but only that one.
A natural question to wonder at this point is: what is the largest sexy prime? The largest we know of has over 11 thousand digits, but is it the largest? Well, we know there are infinitely many prime numbers – Euclid proved this in ancient Greece – but the higher up the list of numbers you get, the rarer prime numbers are. The question of whether there are infinitely many sexy primes depends on the distribution of primes, and this is something we don’t know. In fact, the question of the largest sexy prime leads us neatly to an unsolved problem called Polignac’s conjecture. Alphonse de Polignac, a French mathematician in the 19th Century, made a conjecture in 1849 that, if true, would mean there are infinitely many sexy primes. However, 162 years later, this has neither been proved or refuted.
That mention of the year brings up another connection: 2011 is a sexy prime. That’s because 2017 is also prime. Mathematics is an active field of research. Perhaps by 2017 we’ll have our answer.

Time: 31 minutes. 397 words. Performance: not bad. I discovered my text editor was counting words wrong, so that “we’ve” was counted as two words. Sigh.

N.B.: @stecks suggested that we cover ‘Valentine’s maths’ for today’s episode of the Math/Maths Podcast. @axiomsofchoice suggested sexy primes. So this piece owes a debt to them for the suggestion.

Explanation. More 400 words.