# What does this say about prime numbers?

I was having fun with Sage when I noticed something interesting:

primes = [p for p in range(500) if p in Primes()]
primes_rev = [p for p in reversed(primes)]
mul = map(operator.mul, primes, primes_rev)
sub = map(operator.sub, primes, primes_rev)
div = map(operator.div, primes, primes_rev)


We create a list of prime numbers, primes. We reverse the list, primes_rev. We create new lists from applying math operations to each element of both lists, sum, mul, sub, div. Then we plot the new lists.

list_plot(sum)


list_plot(mul)


list_plot(sub)


list_plot(div)


Does this say anything about prime numbers?

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Interesting pictures, nice to look at and search for meaning. Nowadays, computer experimentation is being used a fair bit to generate plausible conjectures. In this case, you will see if you try that similar plots can be obtained from using most sequences which do not grow too fast. You might for instance repeat with $a_n=n/\log(n)$. The striking picture for mul is basically the general shape of $y=x(N-x)$. The picture for sum is interesting. – André Nicolas Aug 4 '11 at 2:38
I agree with André. The overall trends are just what you'd expect from any sequence of numbers that grows slightly faster than linearly. For example, here's your second plot for if the $n$th entry were $n \log n$; try replacing the multiplication with the other operators and see that you get essentially the same shape as your graphs. – Rahul Aug 4 '11 at 2:47
Speaking of the overall trend, are you aware of the prime number theorem? – Rahul Aug 4 '11 at 2:52
@Rahul Narain: Would you care to answer? I think that (apart from the first sentence!) your comment would be a good answer. – André Nicolas Aug 4 '11 at 3:45
I was aware that different sequence of numbers produce similar plots, but to me the sum and mul plots using prime numbers stand out. – Arlen Aug 4 '11 at 3:56