# Pierre De Fermat and Its Contributions to Number Theory and Probability

## Pierre de Fermat and its contributions to number theory and probability

**Introduction**

Pierre de Fermat was a renowned French lawyer born about 1601-1607 within Beaumont de Lomagne, and consequently died in the year1665 within Castres. He attended the University of Orléans and earned a degree in civil law. Fermat's mathematical work was mainly communicated in terms of letters to associates, frequently with diminutive or no evidence of his underlying theorems (Wilson, 2016). Even though he claimed to have shown his entire arithmetic theorems, few records of his proofs have endured, and numerous mathematicians have doubted some of his claims, particularly granted the complexity of various problems as well as limited mathematical tools accessible to Fermat. Fermat made numerous contributions within numerous locations of mathematics like probability theory, analytic geometry, optics, as well as infinitesimal calculus. Moreover, he is considered to be the inventor of the modern number theory (Beineke & Rosenhouse, 2017). In this paper, I will discuss his contribution to the theory of numbers and corresponding probability.

Fermat’s greatest thoughts are within the area of number theory, which is the purest kind of mathematics entailing the study of whole numbers as well as the associations amidst them. Fermat became predominantly well-known for his Fermat’s Last Theorem.

**Number theory**

The most elegant of the theory was the theorem that each prime in the structure of 4*n* + 1 is exclusively expressible in the form of the summation of two squares. The Fermat numbers denoted in terms of

*F _{n}* = 2

^{2n}+ 1

Where *n* = 0, 1, 2, 3, and 4.

Thus, these numbers are tremendously significant in the learning of the prime numbers as well as Mersenne numbers, which are sturdily studied within number theory (Beineke & Rosenhouse, 2017). Fermat’s lesser theorem is a considerable tool stipulates when p is taken as the prime number and *a* is taken as a certain positive integer, in that case *ap* - *a* is explicitly considered to be divisible by the corresponding p. In arithmetical notation it is represented as

*a ^{p}*

^{-1}= 1(mod

*p*

Fermat never granted demonstrations of his outcomes, and proofs were offered by Gottfried Leibniz, and Leonhard Euler. A further all-purpose theorem pertains to that of a0-(n)-1=0 (mod n), where *a* is deemed to be prime to the* n* and corresponding p(n) is the considered to be the figure of the underlying integers that is less than *n* as well as prime to it.

For intermittent proofs of his theorems, Fermat utilized an infinite descent, which is a reversed structure of interpretation through either recurrence or corresponding mathematical stimulation. It is not acknowledged in case there exist are some primes amongst the Fermat figures for which the actual value of *n* > 5. Moreover, Carl Friedrich Gauss in the year 1796 within Germany established an unforeseen relevance for the Fermat numbers when he proved that a normal polygon possessing *N* sides is considered constructible within a Euclidean logic in case *N* is deemed to be either a prime Fermat figure or a corresponding product of the separate Fermat primes.

The best recognized of Fermat’s numerous theorems in the last theorem. Fermat wrote his chief equation in form of words since he was uninformed of the existing Thomas Harriot’s development regarding symbolic algebra. Utilizing symbols

X^{n}+Y^{n}=Z^{n}

In the case of n=2, then there will be Pythagoras’s theorem, which possesses an infinite number in terms of whole number answers.

Fermat’s Last Theorem asserts that in case *n* is deemed to be a whole number relatively larger than 2, the equation possesses no whole number answers for the x, y, and z. Moreover, Fermat’s verification of his theorem pertaining to the n=4 meant that solely circumstances, where n was deemed to be an odd number, were left to handle (Beineke & Rosenhouse, 2017). Fermat maintained to have proved it for entire values of n, but notably supposed that the margin of his book was too diminutive to write his entire proof.

An odd prime figure typically is denoted as the difference of the existing dual square integers in simply a single way. Thus, Fermat's verification will be as follows.

Taking *n* to be prime, and presume it is equivalent to x^{2} –y^{2} that implies (x+y)(x-y). Through supposition, the simply essential, integral factors pertaining to *n* and *n* as well as unity, therefore x+y=n and corresponding x-y=1. When the equations are solved the solutions will be x=1 /2 (n+1) and corresponding y=1 /2(n-1).

Fermat proved the statement by Diophantus regarding the summation of the dual squares that cannot constitute to 4n-1. Moreover, based on the underlying result which asserts that it is impracticable that the multiplication of a square and corresponding prime structure 4n-1 even if multiplied by a figure that is considered to be prime to the last can be either a square or the summation of dual squares.

In examining the summations of the aliquot sections of numbers, Fermat operated from the underlying Euclid’s solution regarding the perfect numbers that is σ(a) = 2a, where σ(a) indicates the summation of entire divisors of the integer a, incorporating 1 and a, which aids in deriving a complete solution of σ(a) = σ(*b*)= a + b. The general problem is depicted as σ (*a*) = (*p/q*)* a*, plus σ(*x*^{3}) = *y*^{2} and σ(*x*^{2}) = *y*^{3}.

Fermat’s interest regarding the primeness as well as divisibility concluded with a theorem currently fundamental to the theory of congruences, which stipulates that If *p* is considered to be prime and is the least number like* a ^{t} = kp* + 1 for a certain value of k, then t is considered to divides

*p*– 1 (Breedlove & Meltzer, 2017). Regarding the contemporary version, in case p is considered to be prime and p is not capable to divide

*a*, subsequently

*a*

^{p–1}≡ 1 (mod

*p*).

Regarding the result of this very theorem, Fermat examined the divisibility pertaining to the *a ^{k}* ± 1 and thus making his renowned inference that all figures represented as 2

^{2n}+ 1 are all prime except

*n*= 5 that was rejected by Euler (Grant & Kleiner, 2015). In undertaking his study, Fermat actually depended on a widespread accurate control of the powers of the underlying prime figures as well as on the conventional separation of Eratosthenes as an assessment of primeness.

The basic significance of number theory like quadratic residues, as well as quadratic forms actually emanated from Fermat’s learning of the underlying indeterminate equation represented as *x*^{2} – *q* = *my*^{2} for the existing non-square *m*.

Functioning on the underlying theory that any divisor of a figure that is expressed as *a*^{2} + *mb*^{2} ought to be in a certain prescribed format. Moreover, Fermat noted that entire primes in the form 4*k* + 1 can be put in the form of the summation of two squares (Breedlove & Meltzer, 2017).

Moreover, Fermat’s technique regarding the infinite descent was irrelevant solely to the negative suggestions. He revealed that each prime represented as 4*k* + 1 could be depicted as the summation of two squares through rejecting the suggestion for certain prime; obtaining another like prime relatively smaller than the original. In the end, Fermat asserted that this declining succession of the existing primes would produce at the least prime represented as 4*k* + 1 by assumption (Breedlove & Meltzer, 2017). Even though infinite descent is deemed to be unquestionable in its entire logic, its use needs the intelligence of a Fermat because zilch in that logic states the way an individual ought to produce the successive member of the declining sequence for a known problem.

**Probability**

In the year 1654, Blaise Pascal wrote to Fermat recounting gambling problems and Fermat solved the underlying problems in a mathematically rigorous means by examining at the probabilities of entire possible results (Grant & Kleiner, 2015). Thus, Fermat and Pascal are at present renowned as the co-founders of the probability theory. He aided in the development of the probability theory is the mathematics of gambling, risk, as well as change (Beineke & Rosenhouse, 2017). The theory offers the essential principles of the way sporting odds are computed presently from horse racing to corresponding football.

**Conclusion**

In summation, by the year 1662 Fermat had efficiently terminated his occupation as a mathematician. Within his nearly restricted attention regarding the number, a theory established no echo among his existing subordinate contemporaries. Consequently, Fermat more and more returned to the segregation from which he had therefore unexpectedly emerged in the year 1636, and his death in the year 1665 was perceived more like the passing of a splendid elderly man at the expense of a loss to the active technical community.

References

Beineke, J., & Rosenhouse, J. (Eds.). (2017). *The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity* (Vol. 2). Princeton University Press.

Breedlove, B., & Meltzer, M. I. (2017). Extrapolation yields painting, probability, and predictions. *Emerging infectious diseases*, *23*(1), 171.

Grant, H., & Kleiner, I. (2015). Probability: From Games of Chance to an Abstract Theory. In *Turning Points in the History of Mathematics* (pp. 27-35). Birkhäuser, New York, NY.

Turnbull, C. (2017). Probability and Life Contingencies, 1650–1750: The First One Hundred Years. In *A History of British Actuarial Thought* (pp. 1-35). Palgrave Macmillan, Cham.

Wilson, R. (2016). 17th-Century French Mathematics. *The Mathematical Intelligencer*, *38*(1), 96-96.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

**© 2020 Michael Omolo**