4-79-8. James Joseph Sylvester to H. Poincaré

23 June 1893

New College Oxford

My dear M. Poincaré,

I am very glad indeed that you have passed an agreeable time in England and hope that in the last two days at the Athenaeum you found some friends to consort with during your hours of rest from your indefatigable labors. I was obliged to return to Oxford on Friday to attend to some matter of importance here, but would have liked to devote myself more exclusively to you during your residence among us which I shall always look back to with the greatest pleasure as affording me an opportunity of drawing closer an acquaintance which I so highly esteem.

We have found both volumes of your most valuable works on Electricity and Optics for which please to accept my sincere expressions of gratitude.

The work on which I was engaged and in which you were so good as to take an interest has turned out successful and I believe that I may state with certainty as a result of it, that if $n$ exceeds a certain determinable limit (that limit appears to be 2 but can be found by exact investigation), then between $n$ and $3n$ there must lie primes (not necessarily all 4 distinct) of the forms $4n+1$, $4n-1$, $6n+1$, $6n-1$. I hope to be able to extend this result (which is itself an extension of Bertrand’s Postulate slightly modified viz that between $n$ and $2n$ there must be some prime number) to the range of numbers between $n$ and $\mu n$:

Ex. gr. between $n$ and $5n$, I hope to be able to prove that there must lie primes of the forms

$$\left.\begin{array}[]{cccc}10i+1,&3,&7,&9\\ 8i+1,&3,&5,&7\\ 6i+1,&5\end{array}\right\}$$

the last of which conclusions is contained in the previous case.

I beg you to convey to our dear friend Hermite, my grateful acknowledgements for all the trouble he was so kind as to take to obtain signatures to the Memorial addressed to the Authorities of our University press, to induce them to undertake the reprint of my collected works and the extremely handsome letter which I have not seen but am informed he sent over along with the Memorial. I only heard about this yesterday.

“The Delegates of the Press” have decided to comply with the prayer of the Memorial.

M. Hammond desires to be kindly remembered to you and with me hopes that it may not be long before we have again the great pleasure and honor of your company here or in your own country.

Believe me, my dear M. Poincaré, your sincerely attached friend,

J. J. Sylvester

ALS 4p. Collection particulière, Paris 75017.

Time-stamp: "19.06.2023 16:23"

Notes

• 1 Poincaré (1890, 1891).
• 2 Bertrand (1845).