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On some Fermat-quotient congruences

By John Blythe Dobson (j.dobson@uwinnipeg.ca)

Although the author is employed by the University of Winnipeg, he has no affiliation with the Mathematics Department or with any other academic department.

The classical theory of Fermat-quotient congruences of the Eisenstein type, which involve sums of reciprocals of the integers relatively prime to the modulus, was developed for the base 2 over a century ago, notably by Stern (1887), Glaisher (1901), and Lerch (1905). Insofar as such congruences are grouped according to the proportion of the terms in the range from 1 and p − 1 which they comprise, the formulae involving either all the terms, or one-half or one-quarter of them, were essentially exhausted by these authors. However, in the last few decades a resurgence of effort in this area has led to the development of an essentially new group of congruences, involving only one-third or one-sixth of the possible terms. This refinement is of considerable theoretical interest, although such congruences have no value for purposes of actual calculation. This interest persists despite the fact that much of the original motivation for their study was a search for a proof of the first case of Fermat’s Last Theorem, since laid to rest by Andrew Wiles.

The three congruences featured here are all drawn from lengthy papers involving deep arguments and diverse subject-matter, and perhaps have not reached a wide audience. In this note, we shall prove — merely by reference to a few of the classical formulae, and with the aid of some elementary series manipulations in the manner of the nineteenth-century writers — the perhaps surprising fact that all three congruences are equivalent. However, our intention is not in the least to diminish the originality of these results, which were discovered by quite subtle methods, and which in some cases were only presented as corollaries illustrative of much more general theorems. Moreover, without foreknowledge of them it is doubtful whether our proofs would have been found.

We employ the brief version of the Fermat-quotient notation qr to refer to (rp − 1 − 1)/p. All congruences are to be regarded as mod p unless otherwise stated, and throughout we assume (as is typical in the source literature) that p > 3. The three results which will be the main object of our interest have r = 2; we present them in the order of their discovery. The earliest was published in Zhi-Hong Sun (1992), p. 230, Corollary 1.3:

q2 ½
[p/3]
sigma
1
(−1)k − 1 1/k.
(1)

The second is from Dilcher & Skula (1995), p. 389, equation 10.12:

q2 ½
[p/3]
sigma
[p/6] + 1
1/k.
(2)

Most recently, Zhi-Wei Sun (2002), p. 138, published the following formula, in our statement of which we have slightly altered the author’s notation for ease of comparison with earlier work:

q2 −2(−1)(p − 1)/2
[(p + 1)/3]
sigma
1
k odd
(−1)(k − 1)/2 1/k.
(3)

The congruence (1) may be seen as an elegant sharpening of the famous result of Eisenstein (1850) which inaugurated the development of Fermat-quotient formulae as sums of reciprocals:

q2 ½
p − 1
sigma
1
(−1)k−1 1/k.
(4)

Clearly, (1) may be subtracted from (4) to give

p − 1
sigma
[p/3] + 1
(−1)k−1 1/k ≡ 0.
(5)

Next, we may separate the terms into odd and even k, and change only the even k to −k, leading to two distinct sums both over odd k, the second with its sign reversed:

p − 2
sigma
[p/3] + 1
k odd
1/k +
[2p/3]
sigma
1
k odd
1/k ≡ 0.
(6)

We then rearrange the terms into two sums, one covering the entire range from 1 to p − 2, the other covering the overlap:

p − 2
sigma
1
k odd
1/k +
[2p/3]
sigma
[p/3] + 1
k odd
1/k ≡ 0.
(7)

The first sum, running from 1 to p − 2, is congruent to q2 by a formula of Stern (1887), p. 184; Glaisher (1901), p. 27, also proves this formula, generously attributing (on p. 20) the idea to Sylvester (1861), although in our view Sylvester’s remarks on the subject are hardly more than a hint in this direction. In any case, we therefore have for the second sum:

q2
[2p/3]
sigma
[p/3] + 1
k odd
1/k,
(8)

an expression which involves only 1/6 of the possible terms from 1 to p − 1. Once more setting k = −k to make the terms even, we obtain the same sum occupying the same range, but with the sign reversed:

q2
[2p/3]
sigma
[p/3] + 1
k even
1/k,
(9)

which upon division of the factor ½ out of the summand, becomes (2), a formula which has the advantage that all its terms are contained in a continuous block.

To prove (3) from (1), we begin by rewriting (1) so that the factor ½ is moved into the summand:

q2
[2p/3]
sigma
1
k even
(−1)(k − 2)/2 1/k.
(10)

Next, in the range, we change k to the equivalent p − k:

q2
p − 1
sigma
p − [2p/3]
pk even
(−1)(k − 2)/2 1/k.
(11)

Factorising the power of −1, and simplifying the sum, we have:

q2 (−1)(p − 1)/2
p − 2
sigma
[(p + 1)/3] + 1
k odd
(−1)(k − 1)/2 1/k,
(12)

We have thus arrived at a result comparable to one of Sylvester (1861), p. 230 (with an error), but cast in a better form by Stern (1887), p. 187, which was obtained by a clever but quite elementary method:

q2 2(−1)(p − 1)/2
p − 2
sigma
1
k odd
(−1)(k − 1)/2 1/k.
(13)

Subtracting (12) twice from (13) gives (3). The reader should be able to verify without too much effort that through processes similar to the above, any of (1) through (3) may be derived from any other, and that they are thus all equivalent.

As a coda to our survey of these results involving only q2, we remark that the ratio of 1/3 of the possible terms in (1) is reminiscent of the well-known congruence for q3 due to Lerch (1905), p. 476, equation no. 14, which may be easily deduced from Sylvester (1861) (see his Collected Mathematical Papers, 1:23, where an error noticed by Glaisher is silently corrected), but which Lerch cast in a much better form:

q3 −2/3
[p/3]
sigma
1
1/k
(14)

Adding (1) and (8) and dividing the result by 2 yields the relation:

q2 − ¾ q3
[p/3]
sigma
1
k odd
1/k
(15)

which has already been given by Zhi-Hong Sun (1992), p. 240, Corollary 1.12, Equation 2, or alternatively,

q2 − ¾ q3 −½
(p − 1)/2
sigma
[p/3] + 1
1/k   ½
[2p/3]
sigma
(p + 1)/2
1/k,
(16)

if we should prefer to eliminate the condition on k. Either of these is a close parallel to a relation of Emma Lehmer, p. 358, equation 44:

q2 − ¾ q3 ½
[p/6]
sigma
1
1/k.
(17)

Clearly (15) or (16) might be added to or subtracted from (17) to generate individual expressions (mod p) for q3 and q2, respectively, but the results would not appear to be particularly enlightening and we shall not pursue such an exercise here.

Considering the resemblance between (15) or (17) and various p2 congruences considered by Emma Lehmer, it seems likely that with a more careful analysis, some of the results given herein could be sharpened.

Postscript (added 17 Dec. 2007)

The expectation expressed in the very last sentence of these notes that the results (15) and (17) could be generalized to the modulus p2 has just been fulfilled — less than three months later — in Theorem 3.9 of a new article by Zhi-Hong Sun, “Congruences involving Bernoulli and Euler numbers,” Journal of Number Theory 128 (2008): 280-312, at p. 301.

References

Dilcher, Karl & Ladislav Skula. “A New Criterion for the First Case of Fermat’s Last Theorem,” Mathematics of Computation 64 (1995): 363-392.

Eisenstein, [G.] “Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen abhängen und durch gewisse lineare Funktional-Gleichungen definirt werden,” Berichte … der Königlich Preußischen Akademie der Wissenschaften zu Berlin 15 (1850): 36–42. Available online at http://bibliothek.bbaw.de/bbaw/bibliothek-digital/digitalequellen/schriften/anzeige?band=08-verh/1850. Reprinted in Eisenstein’s Werke, 2:705–711.

Glaisher, J.W.L. “On the Residues of r p−1 to Modulus p2, p3, etc.,” Quarterly Journal of Pure and Applied Mathematics 32 (1901): 1-27.

Lehmer, Emma. “On Congruences involving Bernoulli Numbers and the Quotients of Fermat and Wilson,” Annals of Mathematics 39 (1938): 350-360.

Lerch, M. “Zur Theorie des Fermatschen Quotienten…,” Mathematische Annalen 60 (1905): 471–490. Available online at http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0060&DMDID=dmdlog56.

Stern, M. “Einige Bemerkungen über die Congruenz (r p − r)/pa (mod p),” Journal für die reine und angewandte Mathematik 100 (1887): 182–188. Available online at http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0100&DMDID=dmdlog18.

Sun, Zhi-Hong. “[The] Combinatorial Sum and its Applications in Number Theory, I” (in Chinese), Nanjing University Journal — Mathematical Biquarterly 9 (1992): 227-240. A very full summary in English is available on the author’s website, at http://www.hytc.cn/xsjl/szh/coms1.pdf.

Sun, Zhi-Wei. “On the Sum and Related Congruences,” Israel Journal of Mathematics 128 (2002): 135-156.

Sylvester, J.J. “Sur une propriété des Nombres Premiers qui se ratache au Théorème de Fermat,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 52 (1861): 161–163, 212–214 (correction), 307–308 (addendum), 817 (further correction). Reprinted in Collected Mathematical Papers, 2:229–31 (with the briefer corrections incorporated, and also some silent editorial corrections), 232–233, 234–35, 241.


First published 7 October 2007, with minor revisions through 1 June 2009
Postscript added 17 December 2007