4-79-5. James Joseph Sylvester to H. Poincaré

1 Juillet 1891

My dear M. Poincaré,

I thank you very much for your note. It would be very agreable to me to join you at the sea-side at a later period if you intend to prolong your séjour at the sea-side into the month of August.

As to my theorem, about which you inquired, it follows precisely the track of demonstration which I had anticipated when you were with me and both the “intermediate theorems” are correct.

I will now state the results as far as I have proceeded in the inquiry.

Call $ax+b$ when $a$ is prime to $b$ an irreducible linear form to modulus $a$.

Then I say that by taking $n$ suffciently great when $p$ (a prime number) is given, between $n$ and $pn$ must be included prime numbers belonging to every irreducible linear form whose modulus is $2p$. This is certain.

But much more I believe that I am en état to demonstrate that between $n$ and $pn$ must be included prime numbers belonging to every irreducible linear form whose modulus has a totient less than $p$: (by the totient of a number meaning the number of numbers not greater than and prime to it). This is certainly true for the moduli 4, 8, 12 and I think can be proved without difficulty for all other moduli.

Thus ex. gr. between $n$ and $5n$ (when $n$ is suffciently great) must be included prime numbers belonging to each of the forms

$$\begin{array}[]{cccc}8x+1,&3,&5,&7\\ 12x+1,&5,&7,&11\\ 10x+1,&3,&7,&9\\ \end{array}$$

Moreover an inferior functional limit can be found to the number of such primes of each class which will have for its asymptotic value a fractional multiple of $\frac{n}{\log n}$. I believe I can prove that the asymptotic value of the Reciprocals of the primes up [to] $\frac{1}{n}$ is $\log\cdot\log n$: do you know if Tchebycheff has proved this? Believe me,

Your sincerely attached friend,

J. J. Sylvester

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "30.10.2021 19:59"

Notes

• 1 Sylvester invited Poincaré to visit him at New College on 12 June 1891; see Sylvester’s letter to Arthur Cayley, 11 June 1891, in Parshall (1998, 280).
• 2 A note on page 9 of Sylvester (1891) suggests he relied on the presentation by Serret (1879) of Chebyshev’s demonstration of Bertrand’s postulate (Chebyshev, 1852).