With these words Dirichlet begins his attack on a fifty year old conjecture of Legendre.

Conjecture 106.1. if a and d are coprirne positive integers then the arithmetical progression a, a + d, a + 2d,… contains an infinity of primes. (Thus if, for example, we take a = 3, d = 8 then the conjecture states that there are an infinity of primes of the form 8b + 3. The condition that a and d be coprirne is clearly necessary since each a + nd is divisible by the highest common factor of a and d.)

(Fourier seems to have thought such researches a waste of talent. There is a well known letter from Jacobi to Legendre (Jacobi, Collected Works, Vol. I, p. 454) in which Jacobi complains that Poisson had repeated in a report a remark of Fourier where he ‘reproaches Abel and myself for not having given priority to our research in the theory of heat conduction’.

‘Yesterday was my 21st birthday, at that age Newton and Pascal had [already] acquired many claims to immortality.’ Fourteen years after he wrote the postscript above, Fourier was prefect of Isère but had still no claim to the immortality he had craved as a young man. The work on the zeros of algebraic polynomials which he had pursued since his time at Auxerre still gains him a footnote in algebra textbooks – but a footnote is not immortality. His lecturing while at the Ecole Polytechnique had been much praised – but a good lecture is the most ephemeral of triumphs.

However, in 1804 he took up the subject of the propagation of heat. The field of Newtonian mechanics had already been worked over by several masters but the physics of heat, light, electricity and magnetism had still to be brought under the rule of mathematics. The choice of one of these fields thus requires no explanation. The particular choice of heat may have been connected with an obsessional need for warmth which Fourier acquired in Egypt.

In three remarkable years Fourier found the fundamental equations for heat conduction, developed new methods to solve them, applied these methods in a wide variety of cases and produced experimental evidence to support his solutions. At the end of 1807 he had submitted a memoir containing this work to the Academy. A commission consisting of Lagrange, Laplace, Monge and Lacroix was set up to examine it.

In Chapter 100 we discussed the properties which might be demanded from a family Σ of secret codes. In this chapter we describe the code invented by Rivest, Shamir and Adleman.

We start with two very large primes p and q. Write Ν = pq. In the notation of Chapter 100 we take

U=V= {u∈ℤ:0≤u≤N−1}.

We choose a coprirne to (p−1)(q−1) and define T:U→V by

T(u) = uamod N.

How can we recover u from T(u) knowing p and q?

Lemma 102.1. If p and q are prime then φ(pq) = (p−l)(q−1).

Proof. Any integer with a factor in common with pq must be divisible by p and/or q. Thus the only integers m with 1 ≤ m ≤ pq − 1 having a factor in common with pq are p, 2p,…,(q− 1)p and q, 2q,…,(p− 1)q. Since these are all distinct, we know that there are exactly (p−l) + (g−1) integers m with 1 ≤ m ≤ pq − 1 having a factor in common with pq. Hence

φ(pq) = (Pq−1) − ((q−1) + (p−1)) = (p−1)(q−1).

By Theorem 101.2 it follows that

u(P−1)(q−1) ≡ 1 mod N,

and so

ut= u mod N

whenever t ≡ 1 mod (p−1)(q−1).