So is it true that is prime when, and only when, n is a power of 2? Let’s draw up another table:
Table 3. Numbers of the form
It certainly seems as though this is the case.
We first show that if is prime, then n must be a power of 2. To do this, we note that if n had an odd factor (other than 1), then would be composite. For, if we write , where m is odd, then it can be shown that is a divisor of : for example, if , , and , then
It follows that if is prime, then n can have no odd factor, and so must be a power of 2.
Although Fermat was convinced that if n is a power of 2, then must be prime, he confessed that he was unable to prove this, or even to show that the next one, the ten-digit number
is prime. In desperation, he wrote around to fellow mathematicians challenging them to prove his ‘distinguished theorem’. At the time, no-one could do so.
Euler learned of Fermat’s challenge in 1729, and after much tedious experimentation he showed it to be divisible by 641—in fact,
He later discovered that if had any prime factors, then they would have to be of the form , for some integer k. This limits the search considerably, because the smallest prime numbers of this type are 193, 257, 449, and 557 (none of which divides ), and 641 (which does, as we’ve seen).
Is the next Fermat number prime? This number is
Euler showed that any prime factors of must be of the form , for some integer k. Unfortunately, he could find no such prime number that divides , and he eventually gave up his search. But it turns out that is indeed composite, with 274,177 as its smallest prime factor, so it’s not surprising that Euler missed it. In fact,
Worse was to come: the next twenty-six Fermat numbers were also found to be composite. Indeed, no-one has ever found another Fermat number that’s prime, and so Fermat’s conjecture has turned out to be a rather unfortunate one.A geometrical digression
Fermat primes turn up in unexpected places. A celebrated example from geometry arises in the construction of regular polygons—those polygons whose sides all have the same length and whose angles are all the same, such as an equilateral triangle, a square, or a regular pentagon (see Figure 20).
20. Some regular polygons.
The Ancient Greeks were interested in the construction of geometrical figures using only a ruler (for drawing straight lines) and compasses (for drawing circles); the ruler is assumed to be unmarked and no measuring is allowed. Indeed, the very first proposition in Euclid’s Elements, Book I, includes the construction of an equilateral triangle. Briefly it goes as follows (see Figure 21):
21. Constructing an equilateral triangle.
Given a line segment AB, use the compasses to draw the circle with centre A and radius AB, and the circle with centre B and radius BA.
These circles intersect at the point C: draw the lines AC and BC.
Then ABC is an equilateral triangle.
Euclid also showed how to construct squares and regular pentagons, and in Book IV he combined the constructions for triangles and pentagons to produce a regular polygon with sides. He also explained how to construct a regular polygon with 2n sides from one with n sides, by joining the centre of the surrounding circle to the midpoints of its sides (see Figure 22 for the case ).
22. Doubling the number of sides of a regular polygon.
It follows that:
from an equilateral triangle (with 3 sides), we can construct regular polygons with 6, 12, 24, … sides,
from a square (with 4 sides), we can construct regular polygons with 8, 16, 32, … sides,
from a regular pentagon (with 5 sides), we can construct regular polygons with 10, 20, 40, … sides.
So we can already construct regular polygons with 3, 4, 5, 6, 8, 10, 12, 15, and 16 sides.
But no-one has been able to construct regular polygons with 7, 9, 11, 13, or 14 sides, and this leads to the following question:
Which regular polygons can be constructed with an unmarked ruler and compasses?
The breakthrough was provided by the 18-year-old Carl Friedrich Gauss, who discovered a ruler-and-compasses method for constructing a regular 17-sided polygon. He then gave the following answer:
A regular polygon with n sides can be constructed if and only if n is a power of 2 multiplied by unequal Fermat primes.
It follows that one can construct regular polygons with
but not with
The following list gives the number of sides (up to 100) of regular polygons that can be constructed by an unmarked ruler and compasses:
It seems remarkable that Fermat primes arise in such a geometrical problem.Two weird results
We conclude this chapter with two unexpected and rather bizarre results concerning primes.
The first of these is due to W. H. Mills who proved the following startling result in 1947:
There is a number x, with the property that, if we raise it to the powers 3, 9, 27, 81, … (the powers of 3) and then ignore the fractional part, we always get a prime number.
The smallest such number x, sometimes called Mills’ constant, is approximately equal to 1.30637788: for example,
Our second bizarre result involves the idea of a polynomial. This is an expression obtained by adding and subtracting multiples of powers of the variables a, b, c, etc.: for example,
In 1976
