Graciano Ricalde and L'Intermédiaire des mathématiciens

Graciano Ricalde was a Mexican mathematician who was learning advanced mathematics at the end of the 19th century when there were not any people in Mexico who were teaching higher level mathematics. He made use of the French journal L'Intermédiaire des mathématiciens which had been established in January 1894 by Émile Lemoine and Charles-Ange Laisant. Before looking at Ricalde's contributions to the journal, let us give some details of the journal by quoting from the Preface to Part 1 of Volume 1. The reader will see from this information what a marvellous opportunity this journal offered to the young scholar who had nobody in his own country to help him.

Preface to Part 1 of Volume 1 of L'Intermédiaire des mathématiciens.

At the beginning of the first issue of this Journal, we believe it is essential to give the following explanations. L'Intermédiaire des Mathématiciens has nothing in common with the mathematical journals that exist today in France and abroad. We even believe that it does not come close to any previous publication. The title we have chosen was inspired by our reading of L'Intermédiaire des chercheurs et des curieux, a well-known publication which has rendered indisputable services in an intellectual field very different from that which we have in view.

Our main goal is to provide people who usually cultivate Mathematics, or who are interested in it, information on subjects relating to their studies, solutions to questions asked, or bibliographical indications.

Everyone knows how large the number of people who deal with Mathematics has become today, either by profession or by taste. Everyone also knows how much mathematical science has been subdivided into multiple branches and enriched with results.

From this extreme diversity has resulted the obligation, for most of those who study it, to specialise; consequently we are often unaware of what is going on and what is being done in a branch close to that with which we are particularly concerned; also a question, the solution of which one would need, can be very difficult for the one who wishes this solution, or else would require on his part a long research and a great loss of time, while another person considers it, and with good reason in his view, as quite simple.

To bring the two persons in question into contact is therefore to render service to science and to contribute to its progress by sparing useless efforts.

It is this role of the Intermédiaire that we want to take on. For this purpose, we will give access to all the questions that will be asked to us, relating to the Mathematical Sciences, from the most elementary to the most advanced. It often happens that these questions will consist of simple requests for bibliographical information, or relate to simple results of easy but time-consuming calculations.

Whatever the nature of the question, it will be open to the correspondent who sent it to us to have it published under his name, or else to remain anonymous, or even to take a pseudonym of his choice. In this case, it will be enough for him to express the wish, in the covering letter addressed to one or the other of the editors.

In almost all mathematical journals, one finds proposed questions; Usually, whoever asks them has a solution. Here, the order of ideas is quite different. In general, whoever asks a question will only ask it precisely because he lacks the solution, and with the aim of obtaining either this solution or indications relating to it. Often too, the aim will be to quickly obtain a result that one cannot obtain oneself without long work.

In the 17th century, scholars challenged each other and hid their methods from each other; Science has largely profited from this emulation. Today, conditions have changed; Science has spread; the discoveries of each are divulged at the time by the will of the inventors themselves; a kind of collective effort has replaced the individual effort of our fathers, and it is this collective effort that we want to develop further by saving the time spent on research already done, by the quite new means that this journal offers.

The second part of our Collection will be devoted to the answers. Sometimes these answers will be more or less developed solutions, never departing from the corresponding question; sometimes they will be reduced to quite summary information. But, in no case, will we depart from this framework; and we will publish neither articles, nor memoirs, nor even simple notes on subjects extraneous to the questions. This would, in fact, be a duplication with one or other of the excellent mathematical journals which exist in very large numbers. It will not prevent us, on occasion, and on a purely personal basis, from giving our correspondents the information they request from us for the publication of their work, when this is possible for us.

A sample of the questions posed by Graciano Ricalde.

Volume 2 (1895)
2.1. If: a=10m1,b=10n1;nma = 10^{m} - 1, b = 10^{n} - 1; n ≥ m, the following theorem, verified for m=1m = 1 and n=1,2,3,4n = 1, 2, 3, 4; for m=2m = 2 and n=2,3n = 2, 3, etc is it true?

The period of the decimal fraction equivalent to 1a×b\Large\frac 1 {a \times b} consists of a×na \times n digits.
Volume 3 (1896)
3.1. I would be interested in knowing if the number 189 431 482 030 921 is prime or composite. Could a correspondent provide me with this information?

[A reply from Henri Brocard (and several others) says it is actually divisible by 13. Ricalde responds that he knew this but wanted a complete decomposition. He says that he has now found the number is 13 × 14447591 × 1008587 but does not know if the last two numbers are prime. Now, in the 21st century, answering such questions is easy: 14447591 = 2267 × 6373 while 1008587 is prime.]

3.2. I ask for the demonstration of the truth of the following proposition or that of its falsity:
if pp is a prime number, we always have (p1)p1(p+1)(p - 1)^{p-1} - (p + 1) a multiple of p3p^{3}.

3.3. How can we determine all the numbers for which ϕ(N)\phi(N) has a given value, where ϕ(N)\phi(N) is the number of positive integers up to a given integer NN that are relatively prime to NN.

3.4. I would be very interested in knowing if the equation

          x52sx3+s2x+p=0x^{5} - 2s x^{3} + s^{2} x + p = 0

is algebraically soluble or not, and, if it is, to know its roots.

3.5. Is there, for the equation 10x=1+9yz10^{x} = 1 + 9y^{z}, another integer solution other than x=y=z=1x = y = z = 1?
Volume 4 (1897)
4.1. Show that if ... n+1 = multiple of pp, pp being a prime number, then nn will be a divisor of p1p-1.

4.2. Show that if ... n+1 = multiple of NN, NN being some number, then nn will be divisor of the number ϕ(N)\phi(N) which is the number of positive integers up to a given integer NN that are relatively prime to NN.

4.3. Is there any relationship between the smallest value of n, in the preceding questions, and the Gaussian or exponent to which a given number belongs with respect to p or N.
Volume 5 (1898)
5.1. Determine three rational numbers such that their sum, the sum of their two-by-two products and their product are all the squares of three rational numbers.

5.2. Can it be shown that the equation

         x(x+4)(x+6)=y2x(x+4)(x+6) = y^{2}

is impossible with rational numbers?

5.3. Is the following proposition true or not? And if not, how should it be changed?

If NN belongs to the exponent m>1m > 1 with respect to aa, and to the exponent n>1n > 1 with respect to bb, and if we represent by α\alpha the great common divisor of aa and of bb, and by β\beta the least common multiple of mm and nn, the same number NN belongs to the exponent α×β\alpha \times \beta with respect to the divisor α×β\alpha \times \beta, except when, aa being equal to bb, we have

          Nm=1+N^{m}= 1 + multiple of a2a^{2}.

5.4. Charles Bioche (1859-1949) posed the following question in Volume 1 (1894):

Construct a triangle mnpmnp given the feet a,b,ca, b, c of its interior angle bisectors.

Ricalde presents a beautiful solution to this problem which is published in Volume 5 (1898).

5.5. We also learn, from what Ricalde writes in one of the problems, that he has been reading Traité de Mécanique rationnelle Volume 2 Dynamique des systémes - Mécanique analytique (1896) by Paul Appell.

Last Updated June 2023