## 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.^{1}^{1}endnote:
^{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).

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?^{2}^{2}endnote:
^{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).
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).

## References

- Mémoire sur les nombres premiers. Journal de mathématiques pures et appliquées 17, pp. 366–390. link1 Cited by: endnote 2.
- James Joseph Sylvester: Life and Work in Letters. Oxford University Press, Oxford. Cited by: endnote 1.
- Cours d’algèbre supérieure, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 2.
- On arithmetical series (I). Messenger of Mathematics 21, pp. 1–19. link1 Cited by: endnote 2.