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Probabilistic Number Theory Project Description Prerequisites Warm up Problems Hints Resources Teaching Notes Extension Problems Results

Hints for the
Probabilistic Number Theory Problem

These problems all involve picking an integer (or a pair of integers) at random and seeing if it meets a given criterion. Dealing with probability in the integers is hard to think about, because the integers form an infinite set. Well, one way to deal with this is to pick a number at random between 1 and some big number n and see how often it meets the criterion.

Perfect Squares

For perfect squares, you can do a direct estimate. For example, how many perfect squares are there between 1 and 10041? Well, every perfect square in this range is the square of an integer between 1 and  V~ ----
 10041  ~~ 100. And, if you pick a number between 1 and 100 and square it, you get a perfect square between 1 and 10041. So, there are 100 perfect squares between 1 and 10041. So, the probability of a number less than 10041 being a perfect square is
-100-  ~~  .01
Try this reasoning with 10041 replaced by n. Then let n -->  oo .


Now, for the “primality” problem, making this estimate is quite hard. But one way to get a feel for the problem is to try a simulation. For example, you could perform a number of “trials.” In each trial, you can pick a number at random between, say, 1 and 10041. If it’s prime, count it as “good.” Otherwise, don’t count it at all. Do this, say, 1000 times. A good estimate for the probability of a number less than 10041 being prime is the number of good trials divided by 1000.

Of course, you’ll need a computer or calculator for this. Mathematica is especially convenient, because it has a prime-tester PrimeQ built in (actually, all you can conclude if PrimeQ[n] returns True is that there’s a very high probability that n is prime). You can use this to write a function that returns 1 if its input is prime and 0 if it isn’t. Then you can sum this function between 1 and n to get the number of primes less than or equal to n.

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