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At Right Angles | Vol. 3, No. 1, March 2014 15

feature

Convergent and divergent series

Say you have a set S of numbers. You want to know whether there

are finitely many elements in S, or infinitely many. How may we

do this? Here is a possible strategy: Add up all the numbers in

S. If the sum if infinite, then surely S must have infinitely many

elements!

Note that this strategy works only in one direction: If the sum

is infinite, then S has infinitely many elements. But if the sum is

finite, we cannot say anything about the size of S. This strange

situation at one time in history looked impossible, and all kinds of

paradoxes arose because of that, like Zeno’s paradox. But it is easy

There are Infinitely

Many Primes – II

But how many proofs of this?

In Part–I of this article we dwelt on various proofs that show the infinitude of the

primes. These were mostly based on Euclid’s proof — the one for which G H Hardy had

such high praise. All of these start by assuming that there exists a ‘last prime’. Then in

a clever way they construct a number whose prime factors exceed this last prime. The

one proof discussed which does not belong to this category is Pólya’s; he makes use of

the Fermat numbers. The first proof of a completely different nature is Euler’s; he shows

that the sum of the reciprocals of the primes is infinite, and hence there must exist

infinitely many primes. In Part–II we dwell on this proof.

V G Tikekar

Musing on the primes

Keywords: Prime, composite, infinite, Euler, fundamental theorem of arithmetic,

divergent series