3. Historical problems
3.1. Introdution and prehistory
And since the normal methods, which are explained in the books, were not sufficient to prove such difficult propositions, finally I found an absolutely particular procedure for overcoming these difficulties. I called this way of demonstrating descente infinie or indéfinie [infinite or indefinite descent]. At the beginning I used this method for demonstrating negative propositions, like, for example: “there is no number of the form 3n-1 which is equal to a square plus the triple of a square ”; “there is no Pythagorean triangle of which the area is the square of an integer ”. The proof runs by apagoghè eis adunaton in this manner: If a Pythagorean triangle had the area equal to the square of an integer, another triangle less than the first would have the same property. But, if there were a second triangle less than the first with such property, there would be - according to the same reasoning - a third triangle less than the second with the same property, and so on a fourth, a fifth triangle, descending infinitely (à l’infini). But, given a number, there is not an infinite number of numbers less than it (I am speaking about integer numbers). Hence, one deduces the impossibility that a Pythagorean triangle has the area equal to the square of an integer1.

With these words by Fermat, it is clarified (even if not properly defined) what the infinite or indefinite descent is and the history of this method begins exactely with Fermat. The method is so conceived: let us suppose that we have to prove a theorem T and let us reason ad absurdum, posing that T is false and ¬T true. If this position implies that, given two whole positive numbers n and m, n<m, an infinite number of whole numbers would exist between n and m, this is absurd, hence ¬T is false and therefore T is true. Fermat judged himself the inventor of this method because he clearly separated for the first time this procedure from other methods and because he applied the descent to difficult theorems. By other words: he fully understood the potential of this method and thought to make it one of the most important - if not even the most important - in number theory. However, a prehistory of this method exists2. In this period only some isolated and (at least for us) easy theorems were demonstrated by indefinite descent. Among these theorems, we remind the reader the first which was surely proved through this procedure: Euclid in Elements, VII, 31 wants to prove that every composite number has a prime number as a divisor. He reasons this way: let p be a composite number, then p has at least two divisors p 1 and p 2 that are different from p and from 1. If one of the two divisors is a prime number, the theorem is true. So both p 1 and p 2 are composite numbers. Let us consider p 1 . This number will have two divisors p '1 and p '' , 1 and, obviously both p '1 and p '' are smaller than p. If one of these two numbers is a prime number, then the 1 theorem is true. If we suppose that neither p '1 nor p '' are prime, then, considering for example p '1 , we 1 will have two other numbers, for example p ''' and p '''' , that are divisors of p '1 , and p''' < p '1 <p. But if we 1 1 1 deny that p has a prime number as a divisor, this reduction can be continued infinitely and we would be obliged to admit the existence of an infinite number of integers between 1 and p. This is clearly absurd, therefore, the theorem is true. Fermat claimed that “the standard form” of the indefinite descent works when the method is applied to “negative propositions”, namely propositions posed in a non-existencial form; but the application to “positive propositions” needs “the addition of some new principles”3. From the modern point of view that by Fermat appears a rather strange distinction because it is obvious that every negative assertion can be posed in an affermative one which is logically equivalent and viceversa. However, we will see what Fermat wrote makes mathematical sense. First of all, let us see what are these new principles. In the letter to Huygens, Fermat asserted that one of the most important “affirmative” theorems he demonstrated by descent is the famous proposition that every prime number of the form 4n+1 is the sum of two squares. He wrote:
1

Fermat wrote these words in a letter sent to Huygens through Carcavi in August 1659. The letter is titled Relation des nouvelles découvertes en la science des nombres. It can be consulted in Fermat, Oeuvres, 2, pp. 431-436. The discovery of this letter has an interesting history that is summarized in Bussotti, 2006, p. 5, note 4. 2 A good bibliography on the history of the infinite or indefinite descent can be consulted in Goldstein, 1995 and in Bussotti, 2006. With regard to the descent before Fermat the reader can consulte for example Cassinet, 1980; Genocchi, 1855; Vacca, 1927-28. 3 Fermat, Oeuvres, 2, p. 432.

If an arbitrary prime number of the form 4n+1 were not composed of two squares, there would exist a less prime of the same nature and a third one, descending infinitely (à l’infini) to 5, which is the smallest prime of this type and that should not be composed of two squares. But 5 is the sum of two squares. Hence we deduce through the reductio ad absurdum that every number of this type is composed of two squares4.

Literally interpreted the assertion by Fermat makes no sense because it is impossible in the set of the positive integer numbers to descent infinitely and to reach 5. But in this case, it is not difficult to provide a meaningful intepretation: if we suppose that a prime p of the form 4n+1 would exist which is not the sum of two squares, then this would imply the existence of another number p ' , less than p and non-sum of two squares, after that a number p '' , less than p ' and non-sum of two squares and so on, till reaching 5, number that, according to the construction, should not be the sum of two squares, but 5 is the sum of two squares, this implies an absurdity that arises from supposing p non-sum of two squares, hence p is the sum of two squares. More in general: let us suppose that we must prove the theorem T. T is true for an initial set of values. It is possible to use the expression “T is true for the small numbers”. Now let us suppose that T is false for an arbitrary number n and that we are able to construct an algorithm such that, if T is false for n, then T is false for a number m smaller than n. Let us suppose that the process can be iterated and that the “small numbers” are reached. Obviously T might be false in particular exactely for the “small numbers”. However, for the small numbers T is true, so we have a contradiction and such a contradiction arises because we have supposed

¬¿ T to be true. Therefore T is true for every number. We will call this method ¿

reduction-descent. Thus, in the set of methods which can be called descent, it is possible to distinguish, following Fermat’s implicit indications: A) the indefinite descent in a proper sense where if we suppose false the theorem to prove we have: 1) a number m, less than n; 2) a reduction that starts from n and that, from a formal point of view, can be continued infinitely, but that cannot reach m for particular mathematical reasons which are specified in every single theorem; 3) the absurdity which derives from this situation: an infinite number of integers should subsist between m and n; B) the reduction-descent: 1) the theorem to prove is true for some initial values; 2) if we suppose the theorem false for a value we have a reduction that (and this is the main difference with the standard indefinite descent) reaches the initial values; 3) for these values the theorem should be at the same time true and false, this is absurd, hence the supposition 2) has to be removed. Now we can maybe try to answer the question why Fermat deemed the reduction-descent more suitable for the affirmative these. The answer is exactely that in this case there is a set of initial values for which the theorem is true, therefore a mathematician can be induced to exploit this fact in some way and Fermat did this in the manner we have described. Therefore even if there is no difference from a logical point of view between “negative” and “affirmative propositions” and even if, of course, the standard descent can be applied to affirmative theses and the reduction-descent to negative theses, neverthless the form of the two methods justifies the assertions by Fermat. The logical equivalence is only an aspect of the problem; there are other aspects which are even more important to understand the way in which a mathematician proved a certain proposition or in which he applied a method. These aspects depend on the historical background, on the habits which are typical of mathematics in a certain period, on the linguistic form one prefers to give to a problem or to a set of problems and on the general state of the mathematical language in the period. On the basis of these considerations one is perhaps able to understand and to appreciate the plurality of methods (even logically equivalent methods) the mathemticians developed in the course of the time. By the way, this can enriche our conceptual horizon an be useful to the logical research itself. After having described the two variants of Fermat’s method, it is necessary to see the application to nontrivial problems because it is possible to appreciate a method only if we realize how it really works. With regard to Fermat, he claimed to have demonstrated the following significant propositions through descent: 1) there is no Pythagorean triangle of which the area is equal to the square of an integer; 2) every prime of the form 4n+1 is the sum of two squares; 3) every prime of the form 8n+1 or 8n+3 is the sum of a square and the double of another square; 4) every prime of the form 3n+1 is the sum of a square and the triple of another square; 5) Pell’s equation: the equation x 2 =Ny 2 1 (N integer different from a square) has always whole
4

Ibidem, p. 432.

solutions; 6) every integer is the sum of four squares (this assertion is part of the famous polygonal numbers theorem by Fermat); 7) the equations x 3 y 3 =z 3 and x 4  y 4 =z 4 have no integer solution, apart from those trivial; 8) the equation x 2 2=y 3 has the only solution (5,3); 9) the equation x 2 4= y 3 has the only solutions (2,2) and (11,5). The problem with Fermat is that he left the explicit demonstration of the sole assertion 1), for the rest, he left only some more or less vague indications5. The proof of assertion 1 is not difficult, but rather complicated, therefore we prefer to present the demonstration by Euler of the impossibility to solve in integers the equation x 4  y 4 =z 4 . This provides a clear, even if not too trivial, example of a proposition demonstrated by standard indefinite descent. By the way 1) implies the assertion on the equation x 4  y 4 =z 4 . Like an example of demonstration provided by reduction-descent, we will supply Euler’s proof that every number of the form x 2 y 2 , (x,y)=1 is divided only by numbers of its same form. Euler exploited this proposition to demonstrate that every prime of the form 4n+1 is the sum of two squares.

5

Fermat demonstrated n. 1) in his Observations sur Diophante (observation 45), see, Fermat, Oeuvres 1, p. 340; n. 2) is quoted in many letters and in the Observations. What Fermat wrote in the letter to Huygens is the most “complete” reference to reconstruct his possible demonstration; n. 3) and 4) are quoted in a letter to Pascal in 1654, see Fermat, Oeuvres, 2, pp. 310-314. The so called Pell’s equation (better Fermat’s equation) was proposed by Fermat as a challange to Frenicle and the English mathematicians in the period 1657-1658. It is quoted in many letters by Fermat and in the letter to Huygens in 1659, too; with regard to n. 6), Fermat deemed the polygonal numbers theorem like his most signficant. It is quoted many times, even in the Observations (number 18), and in the letter to Pascal in 1654. In the letter to Huygens, Fermat claimed he proved the proposition for the four squares by descent; n. 7) are two particular cases of Fermat’s last theorem, to which Fermat reffered more than once, see letter to Huygens, too. The general proposition is quoted only once in the Observations (number 2); n. 8) and 9) are mentioned in the letter to Huygens like examples of difficult problems solved by descent.