Erdős’s (not Thue’s) Proof of the Infinity of Primes

(19/10/22 In a comment below,  Bernard has noted the proof should be properly attributed to Paul Erdős, not Thue. Thue’s counting argument is similar in spirit, but not as quick.)

This post has no deeper meaning.1 Its purpose is just to present a very nice argument, which we gave in our talk last week, and which we feel should be better known.

One of the beautiful theorems that every school student should see is the infinity of primes.2 The standard Euclid proof tends to be difficult for students to appreciate, however, since, although the arithmetic is trivial, the argument is typically clouded with the unsettling concepts of infinity and contradiction.3

The following proof by Norwegian mathematician Axel Thue involves counting and a little more arithmetic, but avoids a head-on confrontation with infinity. Instead, Thue provides a direct guarantee of the number of primes up to a certain point. Versions of Thue’s argument can also be found at cut-the-knot and in Hardy and Wright,4 but the following seems cleaner to us. Continue reading “Erdős’s (not Thue’s) Proof of the Infinity of Primes”

Signs of the Times

Our second sabbatical post concerns, well, the reader can decide what it concerns.

Last year, diagnostic quizzes were given to a large class of first year mathematics students at a Victorian tertiary institution. The majority of these students had completed Specialist Mathematics or an equivalent. On average, these would not have been the top Specialist students, nor would they have been the weakest. The results of these quizzes were, let’s say, interesting.

It was notable, for example, that around 2/5 of these students failed to simplify the likes of 81-3/4. And, around 2/3 of the students failed to solve an inequality such as 2 + 4x ≥ x2 + 5. And, around 3/5 of the students failed to correctly evaluate \boldsymbol {\int_0^{\pi} \sin 5x \,{\rm d}x}\, or similar. There were many such notable outcomes.

Most striking for us, however, were questions concerning lists of numbers, such as those displayed above. Students were asked to write the listed numbers in ascending order. And, though a majority of the students answered correctly, about 1/4 of the students did not.

What, then, does it tell us if a quarter of post-Specialist students cannot order a list of common numbers? Is this acceptable? If not, what or whom are we to blame? Will the outcome of the current VCAA review improve things, or will it make matters worse?

Tricky, tricky questions.