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

The period of the decimal fraction equivalent to $\Large\frac 1 {a \times b}$ consists of $a \times n$ digits.

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 $p$ is a prime number, we always have $(p - 1)^{p-1} - (p + 1)$ a multiple of $p^{3}$.

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

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

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

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

3.5. Is there, for the equation $10^{x} = 1 + 9y^{z}$, another integer solution other than $x = y = z = 1$?

4.1. Show that if $1.2.3.4.5. ... n+1$ = multiple of $p$, $p$ being a prime number, then $n$ will be a divisor of $p-1$.

4.2. Show that if $1.2.3.4.5. ... n+1$ = multiple of $N$, $N$ being some number, then $n$ will be divisor of the number $\phi(N)$ which is the number of positive integers up to a given integer $N$ that are relatively prime to $N$.

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.

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

         $x(x+4)(x+6) = y^{2}$

is impossible with rational numbers?

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

If $N$ belongs to the exponent $m > 1$ with respect to $a$, and to the exponent $n > 1$ with respect to $b$, and if we represent by $\alpha$ the great common divisor of $a$ and of $b$, and by $\beta$ the least common multiple of $m$ and $n$, the same number $N$ belongs to the exponent $\alpha \times \beta$ with respect to the divisor $\alpha \times \beta$, except when, $a$ being equal to $b$, we have

          $N^{m}= 1 +$ multiple of $a^{2}$.

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

Construct a triangle $mnp$ given the feet $a, b, c$ 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.