History of Fermat quotients (Fermat-Lerch).
(Historie Fermatových kvocientu (Fermat-Lerch).)

*(Czech)*Zbl 1064.11001
Dějiny Matematiky / History of Mathematics 14. Prague: Prometheus (ISBN 80-7196-190-6). xiv, 108 p., open access (2000).

As the title suggests, the book is aimed at Fermat quotients and related topics in the time period 1635–1910. The author starts with a biography and an outline of scientific contributions of Pierre de Fermat. Then he proceeds to Fermat’s Little Theorem, describes the circumstances of its origin and gives the survey of its proofs – from the historical as well as “idea development” point of view.

The third chapter reminds the most important results of the theory of Fermat quotients from the period 1858–1905. Above all, the works of N. H. Abel, G. Eisenstein, J. Sylvester, M. Stern and D. Mirimanov are discussed in details. Let us only recall that for a given prime \(p\) and an integer \(a\) which is relatively prime to \(p\), the notion Fermat quotient stands for \(q(a)=\left(a^{p-1}-1\right)/p\) which is, due to a consequence of Fermat’s Little Theorem, an integer.

In the fourth chapter the contribution of the Czech mathematician Matyáš Lerch (1860–1922) to the considered theory is described, together with Lerch’s biography and the survey of his works on number theory. More information can be found in author’s book [Matyáš Lerch’s work on number theory (Masaryk. Univ., Brno) (1995; Zbl 0874.11005)].

The remaining sections of this chapter are restricted to the results contained in couple of Lerch’s papers devoted to Fermat quotients: [Math. Ann. 60, 471–490 (1905; JFM 36.0266.03)] and [C. R. 142, 35–38 (1906; JFM 37.0225.02)].

More precisely, the following topics are discussed (\(p\) is supposed to be an odd prime, \(a\) an integer relatively prime to \(p\), in the first two items below it is supposed to be positive): the relation of the Fermat quotient \(q(a)\) and the Wilson quotient \(N\), \[ \sum_{a=1}^{p-1}q(a)\equiv N \!\!\pmod p, \qquad \text{where}\;N=\frac{(p-1)!+1}{p}\,, \] and Lerch’s simple proof of this formula; the expression of Fermat quotients through the sums \(q(a)\equiv\sum_{\nu=1}^{p-1} \frac{1}{\nu a}[\frac{\nu a}{p}] \pmod p\), where \([\frac{\nu a}{p}]\) denotes an integral part of the number \(\frac{\nu a}{p},\) and its consequences; the relations for the sums \(Q_k=\sum_{a=1}^{p-1}a^kq(a), \) where \(k\in {\mathbb Z},\) \(0\leq k<p\); the case of the composed modulus – here the Fermat quotient is defined by the formula \(q(a)=\left(a^{\varphi(m)-1}-1\right)/m,\) where \(\varphi(m)\) stands for Euler function. From these sections it is clear that Lerch’s contribution to the theory of Fermat quotients was very important and his results enriched this theory in a truly significant way.

The fifth chapter concerns the works directly following Lerch’s 1905 paper mentioned above, namely the works of A. Friedmann and J. Tamarkine; K. Koutský; T. Agoh, K. Dilcher and L. Skula. The last chapter is devoted to the famous Fermat’s marginalias written into Bachet’s edition of Diofantos’ Arithmetic from 1621: the decomposition of a prime into the sum of two squares and Fermat’s Last Theorem. The discussion of the latter includes the investigation of some special cases and the reminder of the events of the year 1847 when G. Lamé presented his (erroneous) “general proof” of this theorem.

The book ends with five supplements: biographies of the mathematicians mentioned in the text; Fermat’s letter to Carcavi from the August 1659 containing Fermat’s most important discoveries in the field of number theory; a table containing the prime divisors of the form \(k.2^n+1\) of Fermat numbers \(F_m\) including the method, date and the author of the respective discovery; bibliography and the collection of photographs.

The third chapter reminds the most important results of the theory of Fermat quotients from the period 1858–1905. Above all, the works of N. H. Abel, G. Eisenstein, J. Sylvester, M. Stern and D. Mirimanov are discussed in details. Let us only recall that for a given prime \(p\) and an integer \(a\) which is relatively prime to \(p\), the notion Fermat quotient stands for \(q(a)=\left(a^{p-1}-1\right)/p\) which is, due to a consequence of Fermat’s Little Theorem, an integer.

In the fourth chapter the contribution of the Czech mathematician Matyáš Lerch (1860–1922) to the considered theory is described, together with Lerch’s biography and the survey of his works on number theory. More information can be found in author’s book [Matyáš Lerch’s work on number theory (Masaryk. Univ., Brno) (1995; Zbl 0874.11005)].

The remaining sections of this chapter are restricted to the results contained in couple of Lerch’s papers devoted to Fermat quotients: [Math. Ann. 60, 471–490 (1905; JFM 36.0266.03)] and [C. R. 142, 35–38 (1906; JFM 37.0225.02)].

More precisely, the following topics are discussed (\(p\) is supposed to be an odd prime, \(a\) an integer relatively prime to \(p\), in the first two items below it is supposed to be positive): the relation of the Fermat quotient \(q(a)\) and the Wilson quotient \(N\), \[ \sum_{a=1}^{p-1}q(a)\equiv N \!\!\pmod p, \qquad \text{where}\;N=\frac{(p-1)!+1}{p}\,, \] and Lerch’s simple proof of this formula; the expression of Fermat quotients through the sums \(q(a)\equiv\sum_{\nu=1}^{p-1} \frac{1}{\nu a}[\frac{\nu a}{p}] \pmod p\), where \([\frac{\nu a}{p}]\) denotes an integral part of the number \(\frac{\nu a}{p},\) and its consequences; the relations for the sums \(Q_k=\sum_{a=1}^{p-1}a^kq(a), \) where \(k\in {\mathbb Z},\) \(0\leq k<p\); the case of the composed modulus – here the Fermat quotient is defined by the formula \(q(a)=\left(a^{\varphi(m)-1}-1\right)/m,\) where \(\varphi(m)\) stands for Euler function. From these sections it is clear that Lerch’s contribution to the theory of Fermat quotients was very important and his results enriched this theory in a truly significant way.

The fifth chapter concerns the works directly following Lerch’s 1905 paper mentioned above, namely the works of A. Friedmann and J. Tamarkine; K. Koutský; T. Agoh, K. Dilcher and L. Skula. The last chapter is devoted to the famous Fermat’s marginalias written into Bachet’s edition of Diofantos’ Arithmetic from 1621: the decomposition of a prime into the sum of two squares and Fermat’s Last Theorem. The discussion of the latter includes the investigation of some special cases and the reminder of the events of the year 1847 when G. Lamé presented his (erroneous) “general proof” of this theorem.

The book ends with five supplements: biographies of the mathematicians mentioned in the text; Fermat’s letter to Carcavi from the August 1659 containing Fermat’s most important discoveries in the field of number theory; a table containing the prime divisors of the form \(k.2^n+1\) of Fermat numbers \(F_m\) including the method, date and the author of the respective discovery; bibliography and the collection of photographs.

Reviewer: Magdalena Hyksova (Praha)