I have just received an e-mail from an American sailor about last week's column, in which I explained how you could use prime numbers to play heads or tails on the internet. Given that the Marine was deployed leagues under the sea on a submarine and I was sitting above ground, I thought that he might be challenging me to a game. But not at all. It was a curious property of the primes I was using that intrigued him.

The way the game works is to divide prime numbers into “heads” primes and “tails” primes. The “heads” primes are those that have a remainder of 1 when divided by 4. The “tails” primes are the ones that have remainder 3 on division by 4. Being in a submarine gives you lots of time to think and he had turned his attention to whether my prime number coin might be biased.

He'd decided to count how many “heads” primes and “tails” primes there were. For example, up to 10 there are two tails primes (3 and 7) and one heads prime (5). So more tails than heads. Up to 20 there are two more tails primes (11 and 19) and two more heads primes (13 and 17). So tails are still in the lead. As you keep counting, something rather strange seems to happen. Tails always seems to be in the lead. My US Marine is not the first to notice this apparent bias. It was first spotted by Pafnuty Chebyshev, a 19th-century Russian mathematician, and is called the Chebyshev bias.

So do tails always stay in the lead? Well, no. Eventually the heads primes do pull it back - the first time they take the lead is when you get to the prime 26,861. But they immediately lose it again since the heads prime 26,861 is followed by a tails prime 26,863. The next time heads takes the lead isn't until you hit the prime 616,841, but again the lead is immediately snatched back by tails. Although heads seems to do badly, John Littlewood, a British mathematician, proved in 1914 that heads will take the lead an infinite number of times, despite never holding the lead for very long.

A similarly unexpected phenomenon occurs when you toss a real coin. Most people's intuition would tell them that, with a fair coin, heads and tails should share the lead equally often. But this turns out to be wrong. It is likely that heads or tails will be in the lead for most of the time. But the bias in the prime number coin is even more significant than the apparent bias in a real coin and it is only in the past few decades that mathematicians have truly understood the bias.

Two mathematicians, Michael Rubinstein and Peter Sarnak, proved that my US Marine and Chebyshev were right. They found that, measured in the right way, the tails primes lead the heads primes more than 99 per cent of the time. This is different from a fair coin. With this, although heads or tails will often hold a big lead over a long stretch of tosses, in time these leads will get evened out.

But the Chebyshev bias does not contradict the fact that you can use the primes to play a fair game of coin tossing. If you continue to count the primes, then you will find that the number of heads gets closer and closer to 50 per cent. In fact, the difference between them remains so small as to become negligible, so it won't affect the fairness of the prime number coin.

I'm not sure how long my US Marine's tour of duty is but, if it's long enough, there are lots of other prime number challenges out there that we haven't sorted out. In fact, Rubinstein and Sarnak's analysis works only if one of the big conjectures about prime numbers is true. It's called the Riemann hypothesis, and there is a million-dollar prize for the first person to crack that prime number mystery. So I'll be checking my e-mail over the coming months to see what further inspiration 20,000 leagues under the sea might bring.

Conundrum

Let’s play a game. I’ll keep tossing a coin. As soon as the sequence Heads Heads Tails appears, I’ll pay you £20, unless Tails Heads Heads appears first, in which case you pay me £10. Should you play?

Answer

No. It is three times more likely that tails, heads, heads appears. There are four possibilities for the opening two tosses: heads, heads; heads, tails; tails, heads; tails, tails. In the case of heads, heads, I can't beat you. You just have to wait for a tail to appear. However, in the other three cases you can't beat me. The first occurrence of heads, heads must be preceded by a tail, giving me the win.

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