March 14, 2014 | 1

Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter

The symbol π is overloaded in math: depending on context and capitalization, π could be the constant we all know and love (or hate), a projection, a product, or a function. There’s plenty of stuff to read about the circle constant, so today I’m writing about one of those other π’s.

Today’s π is the prime counting function. (A prime number is a whole number whose only divisors are one and itself. A prime can’t be written as a product of two other whole numbers in an interesting way.) The function π(*x*) is pretty easy to understand: for any positive number *x*, π(*x*) tells you how many primes there are that are less than or equal to the number x. So π(1) is 0 because there are no primes smaller than 2, π(2)=1 because 2 is prime, π(3)=2 because both 2 and 3 are prime, π(4)=2, and so on.

I was thinking about π(*x*) because I recently discovered the Twitter account @_primes_. Every hour on the half hour, it tweets the next prime number. (Details on github.) A few weeks ago, for example, it tweeted this prime, not far from π×10,000.

31397

— Prime Numbers (@_primes_) February 20, 2014

Of course, the first question that popped into my head when I discovered this account was how long we had until Twitter’s 140 character limit would prevent @_primes_ from tweeting the next prime. In other words, how many prime numbers are smaller than 10^{141}? (That’s a 1 followed by 141 zeroes, the first integer that doesn’t fit in a tweet.*) And that’s where the prime counting function comes in. I needed to know the value of π(10^{141}). That’s how many hours it is before @_primes_ has to take its ball and go home.

Off I went to look up π(10^{141}). Done and done. Well, kind of. The only problem is that we don’t actually know what π(10^{141}) is. Big numbers are big, and their bigness is hard to comprehend. It only takes a few characters to write 10^{141}, but it’s an impossibly big number. There are only about 10^{80} atoms in the known universe, so 10^{141} is the number of atoms in 10^{61} universes. All that to say, I shouldn’t have been surprised that we don’t know what π(10^{141}) is. In fact, the largest number *x* for which we know the exact value of π(x) is 10^{24}. It’s a really, really big number but a tiny, tiny fraction of 10^{141}.

Luckily, we have the prime number theorem, which is almost as good as knowing π(10^{141}). The prime number theorem tells us that as *x* gets bigger and bigger, π(x) approaches the ratio *x*/ln *x*, where ln is the natural logarithm, or logarithm base *e*. The estimate gets better as *x* gets larger, so the number 10^{141}/ln 10^{141} is a pretty good estimate of how many primes we can fit on Twitter. And that number is…also impossibly huge. Specifically, it’s about 3×10^{138}, or the number of atoms in 10^{58} universes. Yikes!

Remember when I said that big numbers are big? I’m not very good at reasoning about big numbers. It’s totally obvious to me now that 10^{141}/ln 10^{141} should only be a few orders of magnitude smaller than 10^{141}, but I was surprised when I first saw it. 10^{138} just seemed way too big to me! But it’s true. 10^{138} is quite small compared to 10^{141}.

The calculation allayed my fear that we have only a few years left of delightful tweets by @_primes_. 3×10^{138} hours is about 3×10^{134} years, or 2.5×10^{124} times the age of the universe. In an uncertain world, one thing is sure: you, Twitter, our sun, and I will be long dead before character limits keep @_primes_ from tweeting the next prime. What a relief!

*Oops, 10^{140}, a 1 followed by 140 zeroes, is the first number that won’t fit in a tweet, not 10^{141}. I have left the numbers in the post as is. If you need more precise estimates, you can apply the same analysis to 10^{140} instead of 10^{141}. Don’t worry, it’s still a long time before character limits hinder the @_primes_ bot!

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Ummm… doesn’t 10^141 have 142 digits?

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