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little theoremICTMT5, Klagenfurt University, Austria. NiMpIiIoSSN0aEc2Ig== August 2001John Cosgrave, Mathematics Department, St. Patrick's College, Drumcondra, Dublin 9, IRELAND. email: John.Cosgrave@spd.ie web site: http://www.spd.dcu.ie/johnbcos

For Mark Daly - former colleague and friend - who helped me (and still does) when I was really stuck A thing of beauty is a joy for ever. Its loveliness increases; it will never Pass into nothingness. [John Keats (1795-1821), Endymion (1818)] Fermat's 'little' theorem is one of the jewels of Number Theory, and to mark the NiMpIiQrJSUjdGhH anniversary of Fermat's birth (NiMpIiM8JSN0aEc= August 2001), I offer this talk. My talk is not intended as an introduction to Number Theory, nor indeed even as an introduction to Maple, although in both cases it could serve as such. Indeed in the time available (some 30 minutes) it will be possible for me to cover only a very small selection of the topics listed below. Because of the widely reported work of Andrew Wiles, many non-mathematicians have heard of Fermat's Last Theorem - that NiMvLCYpJSJ4RyUibkciIiIpJSJ5R0YnRigpJSJ6R0Yn has no solutions in non-zero integers x, y and z, for NiUvJSJuRyIiJCIiJSIiJg==, ... - and it is probable that more significant mathematical ideas have been developed in attempting to prove that result than any other single mathematical question. But, is the Last theorem one of any consequence? Has anyone (yet!) found an application for it? Such-and-such is true because of Fermat's Last Theorem?On the other hand his little theorem - which was relatively easy to prove (but how many could create a proof ab initio?) - has a vast range of mathematical consequences, and one major practical application. The purpose of my talk is to identify in a single document a number of those consequences, but I am aware that I have not covered all of them; for example I have not touched upon the topic of Carmichael numbers (even though they are there in the background of Guiga's conjecture in Section 7).

When did this theorem start to be called 'Fermat's little theorem? Who (in English) first called it so?Actually not everyone calls it so. In Vol I [1919] of Dickson's monumental three volume [History of the] Theory of Numbers there is an entire chapter devoted to 'Fermat's and Wilson's Theorems.' Hardy and Wright, Davenport, Nagell, ... , simply use 'Fermat's theorem.' And Sierpinski calls it 'Simple Theorem of Fermat' in his 1964 A Selection of problems in the Theory of Numbers.Of course everyone knows what 'Wilson's theorem' is - since there is only one such theorem (but, no doubt, someone will write and tell me of another!) - but 'Fermat's theorem'? Well there are several claimants: the beautiful result - to name but one - that every prime p, with NiMvJSJwRyIiIg== (mod 4), is representable by NiMvJSJwRywmKiQlImFHIiIjIiIiKiQlImJHRihGKQ== for some (unique; ignoring, of course, change of signs, and interchange) integers a and b, could well claim to be 'Fermat's theorem.'On June NiMpIiM/JSN0aEc= 2001 I sent an email to the Number Theory Mailing List enquiring if anyone could answer the above questions. I received several responses (some public, some private), and I will place them in the Fermat's little theorem section of my web site. Briefly, however, I note that a probable answer is that the 'little' came into English from German, but there was no definitive answer as to who first used 'little.'

According to Dickson (and others; see Bibliography) Fermat first announced his theorem in a letter to Mersenne, June (?), 1640.Fermat's 'little' theorem. Let p be prime, and a be any integer with NiMwJSJhRyIiIQ== (mod p), then NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p) In non-congruence language: let p be any prime, and a be any integer not divisible by p, then NiMpJSJhRywmJSJwRyIiIkYnISIi leaves remainder 1 on division by p. A small hand performed illustration. Let NiMvJSJwRyIiKA== and NiMvJSJhRyIiIw==, then NiMvKSUiYUcsJiUicEciIiJGKCEiIiokIiIjIiIn = NiMvIiNrIiIo*NiMsJiIiKiIiIkYlRiU=.Maple examples (the meaning of each command should be clear).restart;p[1] := nextprime(120);a[1] := 2;57 mod 5; # the remainder 57 leaves on division by 5a[1]^(p[1] - 1);a[1]^(p[1] - 1) mod p[1];p[2] := nextprime(10^20 + rand());a[2] := rand(); # Maple has a 12-digit random number generator:a[2] mod p[2];a[2]^(p[2] - 1);That expected, and intentional output can be overcome by using the so-called square-and-multiply method which is incorporated into a Maple command by the use of '&' as follows. (The mathematics of that command, the so-called square-and-multiply method - modular exponentiation - is covered in a Maple worksheet at my web site. It forms part of my 3rd year course on Number Theory and Cryptography.)a[2]&^(p[2] - 1) mod p[2];And let's just see the values NiUvJSJhRyIiIyIiJCIiJg==:2&^(p[2] - 1) mod p[2];3&^(p[2] - 1) mod p[2];5&^(p[2] - 1) mod p[2];Anyone who already knows some Number Theory will understand why I didn't check that NiMvKSIiJSwmJiUicEc2IyIiIyIiIkYrISIiRis= (mod NiMmJSJwRzYjIiIj)It is an automatic consequence ofNiMvKSIiIywmJiUicEc2I0YlIiIiRiohIiJGKg== (mod NiMmJSJwRzYjIiIj)as indeedNiMvKSIiJywmJiUicEc2IyIiIyIiIkYrISIiRis= (mod NiMmJSJwRzYjIiIj)is a consequence ofNiMvKSIiIywmJiUicEc2I0YlIiIiRiohIiJGKg== (mod NiMmJSJwRzYjIiIj) and NiMvKSIiJCwmJiUicEc2IyIiIyIiIkYrISIiRis= (mod NiMmJSJwRzYjIiIj)and now consider this:p := 10^80*rand()^2 + 20! + 2*rand() + 127;length(p);Could that p be a prime? Let's see: if p is prime then automatically we have NiMvKSIiIywmJSJwRyIiIkYoISIiRig= (mod p), etc.So, let's calculate;2&^(p - 1) mod p;Ah! So it isn't, since the output was not 1. I hope you realise how powerful that calculation is, for in a microsecond it has determined that the above p is composite.Question. But what if the output had been '1'? What then? Would it follow that p is prime? If you don't already know the answer to that question, then you might like to think about it before you read Section 4 (Primality Testing).Software Note. The above fast computation might appear impressive, but for seriously large numbers - ones with hundreds of thousands, or several million digits - Maple is, quite simply, absolutely useless. For seriously large numbers a user must avail of remarkable public-domain programs like: prime95.exe (George Woltman's GIMPS project) Everyone will know that George's program has found - on desktop computers - the last four record sized Mersenne primes (most earlier ones had been found on CRAY supercomputers), the most recent of which is the only known prime with more than a million digits. proth.exe (Yves Gallot's Proth project) In July 1999 - using Yves' program - I had the good fortune to find the 115130 digit prime 3*NiMsJiokIiIjIidcQ1EiIiJGJ0Yn (the then NiMpIiM1JSN0aEc= largest known prime), and a factor of the (then, and still) largest known composite Fermat number: NiMvJiUiRkc2IyInWkNRLCYpIiIjKiRGKkYnIiIiRixGLA== That prime proved to be a divisor of the generalised Fermat numbers: NiMvJiUjR0ZHNiQiJ1pDUSIiJCwmKUYoKiQiIiNGJyIiIkYtRi0=, NiMvJiUjR0ZHNiQiJ1ZDUSIiJywmKUYoKiQiIiNGJyIiIkYtRi0=, and NiMvJiUjR0ZHNiQiJ1pDUSIjNywmKUYoKiQiIiNGJyIiIkYtRi0= each of which were new records for their type (of which the last two still stand). Prime Form (Chris Nash's original version, based on Yves Gallot's Proth library)NewPGen (Paul Jobling's sieve program, based on Yves Gallot's Proth library)PRP (George Woltman's probabilistic primality program - which actually uses Fermat's little theorem - to be used in conjunction with NewPGen, and Proth.exe). A faster version - PRP2 - also by GW, exploits the recent Intel Pentium IV chip In May 2001 - using a combination of Yves Gallot's proth, Paul Jobling's NewPGen, and George Woltman's PRP - I also had the good fortune to find the 275,977 digit prime 3*NiMsJiokIiIjIidzbiIqIiIiRidGJw== (the then NiMpIiIoJSN0aEc= largest known prime; it may still be, but these prime rankings have a habit of not lasting for a long time!) That prime proved to be a divisor of the generalised Fermat numbers: NiMvJiUjR0ZHNiQiJ3JuIioiIiQsJilGKCokIiIjRiciIiJGLUYt and NiMvJiUjR0ZHNiQiJ3NuIioiIzUsJilGKCokIiIjRiciIiJGLUYt The first of those broke the previous record (above), and the second established a new record for its type. Last month - here in Klagenfurt - Manfred Toplic found the 246767 digit prime 5*NiMsJiokIiIjIidSKD4pIiIiRidGJw== (the NiMpIiIqJSN0aEc= largest known prime), and a divisor of the generalised Fermat number: NiMvJiUjR0ZHNiQiJ1EoPikiIiQsJilGKCokIiIjRiciIiJGLUYt pfgw (the Chris Nash managed program; the 'pf' is 'Prime Form,' the 'g' is 'Gallot,' and the 'w' is 'Woltman.')That list is not complete; one should consult the web sites of Chris Caldwell (exclusively devoted to prime numbers) or Keith Matthews (which covers Number Theory in general). Those two sites are the "cream of the cream."

I refer my reader to Andr\351 Weil (Section 4, Chapter 2, of his Number Theory) or Harold Edwards (Section 1, Chapter 1, of his Fermat's Last Theorem).Fermat's discovery of his little theorem was a direct outcome of his investigations concerning Euclid's theorem on perfect numbers. More specifically it originated from his investigations into the question of the primality or otherwise of (NiMsJikiIiMlIm5HIiIiRichIiI=).On occasion we say things to our students like: I'm not sure how such-and-such first happened, was discovered, but I think it happened like this... For example, I find myself telling my students how I think Euclid could have discovered his famous theorem on perfect numbers. Also, many years ago, I used to (wrongly) tell how I thought Fermat had discovered his 'little' theorem... ; I thought he had found it by thinking about the decimal expansions of rational numbers. In Section 2 I consider decimal and other expansions, and there you will see how anyone could have formulated Fermat's little theorem had he/she simply asked the right questions having investigated decimal expansions of certain rational numbers.Fermat was led to formulate his theorem via Euclid's theorem on perfect numbers: if (NiMsJikiIiMlIm5HIiIiRichIiI=) is prime then NiMqJikiIiMsJiUibkciIiJGKCEiIkYoLCYpRiVGJ0YoRihGKUYo is perfectOne of my Maple delivered public lectures - The recently discovered record Mersenne prime (from 1995, and available at my web site) - treats the subject of abundant, deficient and perfect numbers, the Euclid theorem, etc - but here suffice it to say that before the mid 1450's it was believed that (NiMsJikiIiMlIm5HIiIiRichIiI=) is prime if and only if n itself is prime, a result blown away by the observation that:NiMvLCYqJCIiIyIjNiIiIkYoISIiIiVaPw== = 23*89 ... ( i )Of course it is trivial that (NiMsJikiIiMlIm5HIiIiRichIiI=) is composite if n is composite, and so the real interest is in the Mersenne numbers {NiMmJSJNRzYjJSJwRw==}, where NiMvJiUiTUc2IyUicEcsJikiIiNGJyIiIkYrISIi, p prime. NiMmJSJNRzYjJSJwRw== is prime for NikvJSJwRyIiIyIiJCIiJiIiKCIjOCIjPCIjPg==, ... , and is composite for NiMvJSJwRyIjNg==, ... Before Fermat (1640) nothing had been investigated beyond those p's just shown, and it was Fermat who noted that:NiMvJiUiTUc2IyIjQiwmKiQiIiNGJyIiIkYrISIi = NiMvIigyJylRKSIjWg==*178481 ... ( ii )One would need to be incredibly numerically insensitive not to notice that '47' is so obviously related to '23':NiMvIiNaIiIj*NiMsJiIjQiIiIkYlRiU=Ones eye gets pulled back to the earlier ( i ) and notices exactly the same phenomenon:NiMvIiNCIiIj*NiMsJiIjNiIiIkYlRiU=But a great mistake for anyone to make, looking at this ab initio, would be to rush in and wonder/declare:whenever NiMmJSJNRzYjJSJwRw== is composite, its least prime divisor q satisfies NiMvJSJxRywmKiYiIiMiIiIlInBHRihGKEYoRig=For see what happens with the next prime after 23:restart;p := 29;M[p] := 2^p - 1;ifactor(M[p]);and so what did Fermat notice? Simply that not only is (returning to (i)) NiMvIiNCIiIj*NiMsJiIjNiIiIkYlRiU=, but also NiMvIiMqKSIiKQ==*NiMsJiIjNiIiIkYlRiU=and, with (ii), not only is (returning to (i)) NiMvIiNaIiIj*NiMsJiIjQiIiIkYlRiU=, but alsoNiMvIiciW3kiIiVneA==*NiMsJiIjQiIiIkYlRiU=In short, one might say that Fermat discovered (then conjectured) what one might call the Fermat-Euler theorem on prime divisors of the Mersenne numbers: for prime p every prime divisor q of NiMvJiUiTUc2IyUicEcsJikiIiNGJyIiIkYrISIi satisfies:NiMvJSJxRyIiIg== (mod p)But it wasn't that alone that Fermat noticed, for he went one step further (he didn't have congruence terminology, but in that language this is what he observed):One has NiMvKiQiIiMiIzYiIiI= (mod 23), and so (squaring, cubing, etc) one has:NiUqJCIiIyIjNiokRiQiI0EqJEYkIiNM etc (are all) = 1 (mod 23)One has NiMvKiQiIiMiIzYiIiI= (mod 89), and so (squaring, cubing, etc) one has:NiUqJCIiIyIjNiokRiQiI0EqJEYkIiNM, ... , NiUqJCIiIyIjKSkqJEYkIiMqKiokRiQiJDUi etc (are all) = 1 (mod 89)One also has (looking at case (ii)) NiMvKiQiIiMiI0IiIiI= (mod 47), and so (squaring, cubing, etc) one has:NiUqJCIiIyIjQiokRiQiI1kqJEYkIiNw etc (are all) = 1 (mod 47)One also has (looking at case (ii)) NiMvKiQiIiMiI0IiIiI= (mod 178481), and so (squaring, cubing, etc) one has:NiUqJCIiIyIjQiokRiQiI1kqJEYkIiNw, ... , NiQqJCIiIyInIVt5IiokRiQiJy4meSI=, etc (are all) = 1 (mod 178481)What I am trying to draw attention to is that for each of the above primes p one has NiMvKSIiIyUibUciIiI= (mod p) for various values of m, but that in all cases one of those m's is (NiMsJiUicEciIiJGJSEiIg==). In short, for the primes NiYvJSJwRyIjQiIjKikiI1oiJyJbeSI= one has NiMvKSIiIywmJSJwRyIiIkYoISIiRig= (mod p)But, is this true for other primes? Yes, of course it is (or appears to be), NiMwJSJwRyIiIw==, of course:for r to 15 do [ithprime(r), 2^(ithprime(r)-1), 2^(ithprime(r)-1) mod ithprime(r)] od; # the reason for the '0' in the 1st output is obvious:And what about changing that '2' to NiYiIiQiIiUiIiYiIic=, etc? for r to 15 do [ithprime(r), 3^(ithprime(r)-1), 3^(ithprime(r)-1) mod ithprime(r)] od; # the reason for the '0' in the 2nd output is obvious:etc. In short, one comes to suspect that NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p) providing, of course, that NiMwJSJhRyIiIQ== (mod p), namely Fermat's little theorem.

Here the simple, and obvious point I wish to make is not only does Fermat's little theorem explain, or help one to understand certain well known phenomena in connection with the decimal expansions of rational numbers, but that those very phenomena themselves - with general bases being used (and not just decimal, i.e. base 10) - can lead someone to re-discovering of Fermat's little theorem. In particular, if one were working with young students, then, with proper guidance, they could be led to conjecture Fermat's little theorem.Specifically I refer to the quickly observed fact that the number of digits in the period of the decimal expansion of NiMqJiIiIkYkJSJwRyEiIg== (where p is any prime with NiQwJSJwRyIiIzBGJCIiJg==) appears to an investigator to be (and may be proved using Fermat's little theorem to be):either (NiMsJiUicEciIiJGJSEiIg==) or a divisor of (NiMsJiUicEciIiJGJSEiIg==)Start. Many sensitive young people are fascinated (or, at least, used to be!) with phenomena like:NiMvKiYiIiJGJSIiIyEiIi0lJkZsb2F0RzYkIiImRic=NiMvKiYiIiJGJSIiJCEiIi0lJkZsb2F0RzYkIilMTExMISIp... ad infinitumNiMvKiYiIiJGJSIiJSEiIi0lJkZsb2F0RzYkIiNEISIjNiMvKiYiIiJGJSIiJiEiIi0lJkZsb2F0RzYkIiIjRic=NiMvKiYiIiJGJSIiJyEiIi0lJkZsb2F0RzYkIiltbW07ISIp... ad infinitumNiMvKiYiIiJGJSIiKCEiIi0lJkZsb2F0RzYkIidkRzkhIic= 142857 142857 ... ad infinitum [Here, and elsewhere, I make spaces to emphasise the periodic block.]NiMvKiYiIiMiIiIiIighIiItJSZGbG9hdEc2JCInOWRHISIn 285714 285714 ... ad infinitum and - as almost everyone who has ever investigated such matters (without knowing that it has all already long been discovered) - one quickly finds thatthe rational number NiMqJiUibUciIiIlIm5HISIi (where m and n are positive, NiMyJSJtRyUibkc=) has a periodic decimal expansion provided n is not divisible by 2 or 5NiMvKiYiIiJGJSIjQCEiIi0lJkZsb2F0RzYkIiY+dyUhIic= 047619 047619 (I have deliberately made spaces to emphasise the periodic block) ...otherwise NiMqJiUibUciIiIlIm5HISIi has an eventually periodic decimal expansionNiMvKiYiIiJGJSIjQSEiIi0lJkZsb2F0RzYkIiIhISIj 45 45 45 45 ... (here n is divisible by 2)NiMvKiYiIiJGJSIkJj0hIiItJSZGbG9hdEc2JCIiISEiIw== 054 054 054 ... (here n is divisible by 5) Anyone who gets preoccupied with the lengths of those periodic blocks quickly makes an often made (re)discovery: for prime p (NiQwJSJwRyIiIzBGJCIiJg==) the length of the period of the decimal expansion of NiMqJiIiIkYkJSJwRyEiIg== is either (NiMsJiUicEciIiJGJSEiIg==) or a divisor of (NiMsJiUicEciIiJGJSEiIg==). A sensitive eye gets quickly drawn towards the 'prime' element in all of this because the examples with long periods - long in relation to the size of the denominator - having encountered examples like:NiMvKiYiIiJGJSIiKCEiIi0lJkZsb2F0RzYkIidkRzkhIic= 142857 142857 ... [period length 6]NiMvKiYiIiJGJSIjPCEiIi0lJkZsb2F0RzYkIjBadzYlSE4jKWUhIzs= 0NiMiMFp3NiVITiMpZQ== ... [period length 16]NiMvKiYiIiJGJSIjPiEiIi0lJkZsb2F0RzYkIjJAJW90JSp5OmpfISM9 0NiMiMkAlb3QlKnk6al8= ... [period length 18] and the other primes p, up to 100, for which NiMqJiIiIkYkJSJwRyEiIg== has period length (NiMsJiUicEciIiJGJSEiIg==) are 23, 29, 47, 59, 61 and 97. Anyone who knows sufficient Number Theory will know that they are primes p for which NiMvLSYlJG9yZEc2IyUicEc2IyIjNSwmRigiIiJGLCEiIg==; in other words they are primes for which 10 is a primitive root. [See, too, Section 7 on open problems]. As soon as the eye has got drawn in to NiMqJiIiIkYkJSJwRyEiIg== for NicvJSJwRyIiKCIjPCIjPiIjQiIjSA==, etc, then the eye returns to look at the decimal expansions of the reciprocals of the other primes (not 2 or 5), and notices:NiMvKiYiIiJGJSIiJCEiIi0lJkZsb2F0RzYkRiZGJw== 3 3 3 3 3 ... [period length 1]NiMvKiYiIiJGJSIjNiEiIi0lJkZsb2F0RzYkIiIqISIj 09 09 09 09 ... [period length 2]NiMvKiYiIiJGJSIjOCEiIi0lJkZsb2F0RzYkIiZCcCghIic= 076923 076923 ... [period length 6]NiMvKiYiIiJGJSIjSiEiIi0lJkZsb2F0RzYkIi9IaF5rIWVBJCEjOg== 0NiMiL0hoXmshZUEk ... [period length 15] Maple has a command for computing those periodic decimals expansions, and it's called 'pdexpand'. To access it one needs to load Maple's Number Theory package:with(numtheory);order(10, 7);pdexpand(1/31);This is not a Maple tutorial, but anyone who wishes may consult what Maple has to say about 'pdexpand' by executing the following line (first remove the '#', and then execute):# ?pdexpandpdexpand(135/14);means that NiMvKiYiJE4iIiIiIiM5ISIiLSUmRmxvYXRHNiQiIycqRig= 428571 428571 428571 ... ad infinitum, andconvert(PDEXPAND(-1, 2, [1, 1], [9, 0, 1, 3]), rational);means that NiMsJC0lJkZsb2F0RzYkIiQ2IyEiIyEiIg== 9013 9013 9013 ... = NiMsJComIigsJWY1IiIiIiddKipcISIiRig=.pdexpand(1/7);pdexpand(1/21);pdexpand(1/22);pdexpand(1/185);In passing I cannot resist asking if from: NiMvKiYiIiJGJSIjNiEiIi0lJkZsb2F0RzYkIiIqISIj 09 09 09 ... (A)my reader can determine the decimal expansion of NiMqJiIiIkYkKiQiIzYiIiMhIiI=? In other words, what do you get if you square both sides of (A)?pdexpand(1/11);pdexpand(1/11^2);And NiMqJiIiIkYkKiQiIzYiIiQhIiI= ?pdexpand(1/11^3);A suggestion for playing. Much fun may be had by investigating (and explaining what's going on with) decimal expansions of NiUqJiIiIkYkIiQsIiEiIiomRiRGJCokRiUiIiNGJiomRiRGJCokRiUiIiRGJg==, ... ; NiUqJiIiIkYkIiUsNSEiIiomRiRGJCokRiUiIiNGJiomRiRGJCokRiUiIiRGJg==, ... ; etc. A knowledgeable practitioner should be able to quess, and prove results.A word of warning. One must be careful about saving ones before executing some commands!!Returning to above. And now to observe the obvious connection to Fermat's little theorem. All, I believe, becomes clear from almost one reflection; simply consider the decimal expansion of, say, NiMqJiIiIkYkIiIoISIi:NiMvKiYiIiJGJSIiKCEiIi0lJkZsb2F0RzYkIidkRzkhIic= 142857 142857 ... ad infinitumHow does one prove that the non-terminating decimal on the right hand side is equal to the (rational) number NiMqJiIiIkYkIiIoISIi? Of course one needs to have studied infinite series to give a precise meaning to such an object...It's very simple, and straightforward, providing one knows that:NiMvLSUkU3VtRzYkKSUickclIm5HL0YpOyIiIiUpaW5maW5pdHlHKiZGKEYsLCZGLEYsRighIiJGMA== for NiMyLCQiIiIhIiIlInJH < 1and thus:.111... ad infinitum = NiMvLSUkU3VtRzYkKSomIiIiRikiIzUhIiIlIm5HL0YsO0YpJSlpbmZpbml0eUcqJkYpRikqJkYqRiksJkYpRilGKEYrRilGKw== = NiMvKiYiIiJGJSwmIiM1RiVGJSEiIkYoKiZGJUYlIiIqRig=.010101... ad infinitum = NiMvLSUkU3VtRzYkKSomIiIiRikqJCIjNSIiIyEiIiUibkcvRi47RiklKWluZmluaXR5RyomRilGKSomRitGLCwmRilGKUYoRi1GKUYt = NiMvKiYiIiJGJSwmKiQiIzUiIiNGJUYlISIiRioqJkYlRiUiIyoqRio=.001001001... ad infinitum = NiMvLSUkU3VtRzYkKSomIiIiRikqJCIjNSIiJCEiIiUibkcvRi47RiklKWluZmluaXR5RyomRilGKSomRitGLCwmRilGKUYoRi1GKUYt = NiMvKiYiIiJGJSwmKiQiIzUiIiRGJUYlISIiRioqJkYlRiUiJCoqKkYqetcTwo points, now, are simply these:1. Fermat's little theorem (with NiMvJSJwRyIiKA==, NiMvJSJhRyIjNQ==) forces NiMqJiIiIkYkIiIoISIi to have the decimal (10) expansion that it has.2. The decimal expansion of NiMqJiIiIkYkIiIoISIi, being what it is, forces Fermat's little theorem to hold for NiMvJSJwRyIiKA==, NiMvJSJhRyIjNQ==.Why? It's simple;1. By Fermat's little theorem, with NiMvJSJwRyIiKA== and NiMvJSJhRyIjNQ==, we have NiMvKiQiIzUiIiciIiI= (mod 7), and thus 7 divides (NiMsJiokIiM1IiInIiIiRichIiI=). Performing the division by 7 we find that NiMvLCYqJCIjNSIiJyIiIkYoISIiIiIo*142857, and so it follows that:NiMvKiYiIiJGJSIiKCEiIiomIidkRzlGJSwmKiQiIzUiIidGJUYlRidGJw== = .142857 142857 142857 ... ad infinitum2. If one has determined that the decimal expansion of NiMqJiIiIkYkIiIoISIi is given by: NiMvKiYiIiJGJSIiKCEiIi0lJkZsb2F0RzYkIidkRzkhIic= 142857 142857 ... ad infinitumthen one has NiMvKiYiIiJGJSIiKCEiIiomIidkRzlGJSwmKiQiIzUiIidGJUYlRidGJw==, namely NiMvLCYqJCIjNSIiJyIiIkYoISIiIiIo*142857. Thus 7 divides (NiMsJiokIiM1IiInIiIiRichIiI=), and so it follows that NiMvKiQiIzUiIiciIiI= (mod 7).I hardly need write any more on this?

Here I begin with a famous empirical discovery of Fermat's: every odd prime divisor of (NiMsJiokJSJ4RyIiIyIiIkYnRic=) leaves remainder 1 on division by 4In the following, this is what I am doing: I first form a random NiMsJiokJSJ4RyIiIyIiIkYnRic=, factor it, and then verify that every odd prime divisor leaves remainder 1 on division by 4. Bear in mind thatif x is odd then one of the primes dividing NiMsJiokJSJ4RyIiIyIiIkYnRic= will be 2 itself (and it is trivial that NiMqJCIiI0Yk will not be a factor), otherwise all the other prime factors will be odd (and, in extremis, there might only be one: NiMvLCYqJCIiJCIiIyIiIkYoRihGJw==*5)if x is even then all primes dividing NiMsJiokJSJ4RyIiIyIiIkYnRic= will be odd (and, in extremis, there might only be one: NiMvLCYqJCIiI0YmIiIiRidGJyIiJg==) x := rand();n := x^2 + 1;m := ifactor(n);L := [op(m)];r := nops(L); # how many prime factors there arefor k to r do p[k] := op(L[k]) od;for k to r do p[k] mod 4 od;All those odd primes (which will change every time the two commands are re-executed) are congruent to 1 mod 4. Euler gave a proof based on Fermat's little theorem (see Section 8).An extension of that result is thatevery odd prime divisor of (NiMsJiokJSJ4RyIiJSIiIkYnRic=) leaves remainder 1 on division by 4Here, numerical experimentation like the above (using rand()) would be problematic, since Maple - almost certainly - would have difficulty in performing the resulting factorisations (and, in fact, it is precisely the difficulty of factoring that is the basis of RSA public-key cryptography).Instead, I choose more modestly sized n's to factor:x := rand() mod 1234321; # to reduce the size of 'x':n := x^4 + 1;m := ifactor(n);L := [op(m)];r := nops(L); # how many prime factors there arefor k to r do p[k] := op(L[k]) od;for k to r do p[k] mod 8 # note the change to mod 8 od;That result - re (NiMsJiokJSJ4RyIiJSIiIkYnRic=) - may be proved in the same way as the (NiMsJiokJSJ4RyIiIyIiIkYnRic=) result. In general one has:every odd prime divisor of (NiMsJiklInhHKSIiIyUibUciIiJGKUYp) leaves remainder 1 on division by NiMpIiIjLCYlIm1HIiIiRidGJw==This result enables trial factoring of Fermat numbers to be eased, and there is another simple extension of it that saves a further 50%...Yet another application of Fermat's little theorem is to the trial factoring of Mersenne numbers (NiMvJiUiTUc2IyUicEcsJikiIiNGJyIiIkYrISIi, with p prime): Theorem. Every prime divisor q of NiMmJSJNRzYjJSJwRw== satisfies NiMvJSJxRyIiIg== (mod p).It is pleasing that that Theorem can be proved by using Fermat's little theorem, when it was precisely the very discovery of that theorem which led Fermat to discover his little theorem in the first instance.Comment. A great deal more about Mersenne numbers (including the Lucas-Lehmer test) is available in my 1995 Maple public lecture "The recently discovered record largest known prime," and a great deal more more about Fermat numbers (including the Pepin test) is available in my 1999 Maple public lecture"The history of Fermat numbers from August 1641," both at my web site.

Here I only remark that the modern study of primality testing really begins with the trail blaising work of Lucas, via Fermat's little theorem, and I urge any serious reader to rush to their bookdealer and obtain a copy of the wonderful book by Hugh C. Williams (details in the Section 8 bibliography).I have briefly hinted at the sort of use that can be made of Fermat's little theorem to establish that a number is composite, but a great deal more is involved in using it as a starting point in proving that a number is prime.I refer the interested reader to my web site, where I have many Maple worksheets - in the 2nd and 3rd year, and Public and Other Lectures sections of my site's Maple section - devoted to primality testing.

Frequently it is (correctly) stated that a fundamental element in the renowned RSA cryptographic method is the use of the Fermat-Euler theorem: let n (NiMyIiIiJSJuRw==) be a natural number, and a be any integer with NiMvLSUkZ2NkRzYkJSJhRyUibkciIiI=, then NiMvKSUiYUctJSRwaGlHNiMlIm5HIiIi (mod n), where NiMtJSRwaGlHNiMlIm5H is the Euler phi-function (the number of integers between 1 and n that are relatively prime to n). In fact it is only a very special case of this theorem that is needed for the RSA application, namely the case where NiMvJSJuRyUjcHFH, where p and q are distinct primes (in practical applications p and q are both large, and with some added refinements for security purposes). A fairly detailed exposition of the RSA method may be read in my Maple public lecture - Bill Clinton, Bertie Ahern, and digital signatures - in the Maple Public and Other Lectures section of my web site.The two prime version of Fermat's little theorem is simply this: let p and q be distinct primes (in cryptographic applications they will be large, but not just merely large (as is sometimes incorrectly stated); to see why, refer to Section 6 )a be any integer with NiMwJSJhRyIiIQ== (mod p) and NiMwJSJhRyIiIQ== (mod q)then NiMvKSUiYUcqJiwmJSJwRyIiIkYpISIiRiksJiUicUdGKUYpRipGKUYp (mod pq) One may easily give a proof [see Section 8] of the two prime version which is independent of the normal proof of the full Fermat-Euler result. Here I merely give a Maple numerical illustration:restart;p := nextprime(10^50 + rand()*rand()*rand());q := nextprime(10^60 + rand()^2*rand()^3);a := rand()^2 + 30!;igcd(a, p);igcd(a, q);a&^((p-1)*(q-1)) mod p*q;

In a nutshell, John Pollard's famous (NiMsJiUicEciIiJGJSEiIg==) factorisation method (1974) gives a strategy - based on an ingenious use of Fermat's little theorem (full details are available in the NiMpIiIkJSNyZEc= year Maple section of my web site) - for uncovering a prime factor of an integer n, if that n has a prime factor p such that (NiMsJiUicEciIiJGJSEiIg==) has only relatively small prime factors...And why - apart from its intrinsic interest - might one want to find a factor? Well one need look no further than the renowned RSA cryptographic method (a full treatment of which may be read in my Maple public lecture Bill Clinton, Bertie Ahern, and digital signatures): the basis for its security rests on the current difficulty of the famous factorisation problem, of finding a fast method (or prove there isn't one) for factoring general n. Note. In the following examples, Maple is using the 1971 Brillhart-Morrison continued fraction factorisation method.ifactor(15);n23 := 23!;length(n23);ifactor(n23);ifactor(n23 + 1);n90 := 90!;length(n90);ifactor(n90);But Maple cannot factor (NiMsJi0lKmZhY3RvcmlhbEc2IyIjISoiIiJGKEYo) (and I've placed the 'comment sign', # , before the command to prevent re-execution.# ifactor(n90 + 1);ifactor(n90 + 1, easy);which informs that (NiMsJi0lKmZhY3RvcmlhbEc2IyIjISoiIiJGKEYo) is a composite '139' digit number.Note. That 139 digit number may be proved to be composite (not prime) by using Fermat's little theorem:if (NiMsJi0lKmZhY3RvcmlhbEc2IyIjISoiIiJGKEYo) is prime, one would have NiMvKSUiYUclJG45MEciIiI= (mod NiMsJiUkbjkwRyIiIkYlRiU=)for all integers a not divisible by (NiMsJiUkbjkwRyIiIkYlRiU=). So, testing to see what happens when NiMvJSJhRyIiIw==, one finds:2&^n90 mod (n90 + 1);and so it follows immediately that (NiMsJiUkbjkwRyIiIkYlRiU=) is not prime. (One says that (NiMsJiUkbjkwRyIiIkYlRiU=) has failed the Lucas-Fermat test to the base 2.)Note. On NiMpIiM1JSN0aEc= April this year, Sean Irvine announced to the Number Theory mailing list the complete factorisation of (NiMsJiUkbjkwRyIiIkYlRiU=); it is the product of three primes, NiUlInBHJSJxRyUickc= : p := 1381173038633;q := 42025070023047325132446149666132026718715669472074383;r := 25596421444788555447555320241499448369527283715169223676054601753562783959;is(n90 + 1 = p*q*r);Now here is the famous 129 digit number, RSA129:RSA129 := 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541;length(RSA129);I am certainly NOT going to attempt to factor it!! It first came to public attention in Martin Gardiner's much read column in the August 1997 issue of the Scientific American. Rivest, Shamir and Adleman (of RSA fame) threw this number out as a challenge to be factored. Given the then state of mathematical knowledge (as far as factoring was concerned) and computer power, they estimated that it would take some 20,000 years to factor it, and thereby decrypt a message which they had encrypted using it as the 'n' part of a public modulus.Briefly, a method known as the 'Quadratic Sieve' - introduced in 1981 by the US mathematician Carl Pomerance - together with thousands of computers worldwide (organised by the Dutch mathematician Arjen Lenstra) - factored it by April 1994, and decrypted the RSA-encrypted message (which, incidentally, was: "The magic words are squeamish ossifrage.") Lenstra, and his co-workers found that RSA_129 was the product of the two prime numbers (I show them individually, then their produst, and then show their product really is RSA_129):f1 := 3490529510847650949147849619903898133417764638493387843990820577;length(f1);f2 := 32769132993266709549961988190834461413177642967992942539798288533;length(f2);f1*f2;is(RSA129 = f1*f2);If I were to foolishly enter the Maple command:# ifactor(RSA_129);then the timer would stay on for MANY, MANY years ... .I am going to create a number - which I will call by the name 'much_greater_than_RSA129' - which will have these properties:it will be the product of two primes 'P' and 'Q'I will choose P to be f2 (the greater of the two primes whose product is RSA129)I will choose Q to be much greater than P Those choices will automatically make 'much_greater_than_RSA129' be much greater than RSA129 (hence the name).This is how I will do it:P := f2;To save time in my lecture I neutralise the following command, but beforehand copy and paste the output to the next command line.# Q := nextprime(4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903679999999999999630);Q := 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000001;much_greater_than_RSA129 := P*Q;length(much_greater_than_RSA129);But that MUCH LARGER number can now be factored in seconds by exploiting the Pollard (NiMsJiUicEciIiJGJSEiIg==) factorisation method:Pollard := proc(n) local a,k; a[1]:=2: for k from 2 while igcd(n,a[k-1]-1 mod n)=1 do a[k]:=a[k-1]&^k mod n od; lprint(n,`is the PRODUCT of`, igcd(n, a[k-1]-1 mod n), `and`, n/igcd(n, a[k-1]-1 mod n)) end:Pollard(much_greater_than_RSA129);Wonderful? Yes? Of course I deliberately made up the prime Q so that all the prime factors of (NiMsJiUiUUciIiJGJSEiIg==) were small:ifactor(Q - 1);You may have noticed that I choose Q so that the prime factors of Q are the first twenty-one primes: 2, 3, 5, ... , 71 and 73.Finally, look at the two prime factors - f1 and f2 - of the RSA129 number; see how they behave:ifactor(f1 - 1);ifactor(f2 - 1);Rivest, Shamir and Adleman knew what they were doing when the made up RSA129!!And here you can see why it had been so difficult to find the prime factors q and r of (NiMsJi0lKmZhY3RvcmlhbEc2IyIjISoiIiJGKEYo):ifactor(q-1, easy);ifactor(r-1, easy);

First, recall the meaning of the term primitive root modulo p (p prime): a is said to be a primitive root modulo p if NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p), and NiMwKSUiYUclInJHIiIi (mod p) for all NiUvJSJyRyIiIiIiIyIiJA==, ... , (NiMsJiUicEciIiIiIiMhIiI=). In other words, a is a primitive root modulo p if NiMmJSRvcmRHNiMlInBHNiMvJSJhRywmJSJwRyIiIkYnISIi.Maple examples:restart;with(numtheory): # needed for 'order' and 'phi'p := nextprime(200);order(2, p);order(3, p);order(4, p); # not a surprise:order(5, p);So, 2 and 3 are primitive roots modulo 211, 4 (for obvious reasons, known in advance) is not a primitive root mod 211, and 5 also is not a primitive root modulo 211.Recall a classic theorem of Gauss that p has exactly NiMtJSRwaGlHNiMsJiUicEciIiJGKCEiIg== primitive roots in the interval [1, NiMsJiUicEciIiJGJSEiIg==], and let us make a list of all those primitive roots - and count how many there are - for NiMvJSJwRyIkNiM=:L := []: count := 0: for a to (p-1) do if order(a, p) = p-1 then L := [a, op(L)]; count := count+1; fi od; sort(L); count;To determine how many there are one can, trivially, resort to:nops(L);phi(p-1); # the Gauss result:That's all very straightforward, and well known. But now let us look at the fascinating unknown: Fix the value of a, and ask: for how many primes p is a a primitive root? In other words, for how many primes p does one have:NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p) ... (i)but NiMwKSUiYUclInJHIiIi (mod p), all r in [1, NiMsJiUicEciIiJGJSEiIg==] ... (ii)Here one has the (famous) conjecture of Emil Artin. Let a be any integer (NiMwJSJhRywkIiIiISIi, and not a square), then a is a primitive root mod p for infinitely many primes p. (In 1958 Schinzel and Sierpinski showed Artin's conjecture is a consequence of the prime tuples conjecture, and in 1986 Heath-Brown proved (unconditionally) Artin's conjecture for infinitely many a (but no single 'a' is known for which the conjecture is true).)A remarkable quantitative version (for such an a) of this conjecture - by Artin - had been: Let NiMtJiUjcGlHNiMlImFHRiY= be the number of primes less than or equal to x for which a is a primitive root, then:NiMtJiUjcGlHNiMlImFHNiMlInhH is asymptotic to NiMqKCUiQUciIiIlInhHRiUtJSNsbkc2I0YmISIi as NiNmKjYjJSJ4RzciNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIlKWluZmluaXR5R0YqRipGKg==where A (Artin's constant) is defined by: NiMvJSJBRy0lKFByb2R1Y3RHNiQsJiIiIkYpKiZGKUYpKiYlInBHRiksJkYsRilGKSEiIkYpRi5GLi9GLDsiIiMlKWluZmluaXR5Rw== ( = 0.3739558136... )In other words:NiMqJi0mJSNwaUc2IyUiYUc2IyUieEciIiIqJkYqRistJSNsbkdGKSEiIkYv tends to NiMlIkFH as NiNmKjYjJSJ4RzciNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIlKWluZmluaXR5R0YqRipGKg==[I should point out to readers, perhaps not entirely familiar with Number Theory, that the ' NiMqJiUieEciIiItJSNsbkc2I0YkISIi ' is the main asymptotic term of the famous, classic Prime Number Theorem; it is, roughly, the number of primes up to x.]Above I wrote "had been", because computational work by the Lehmers (D.H. and E., 1962) revealed an error (an independence error in a probability argument), that error being set right by Tate and Lang (both of whom had been Artin's students) in their 1965 preface to Artin's Collected Papers (1965).Hooley (1967) proved that the above quantitative result is true for squarefree a (NiMyIiIiJSJhRw==) subject to the Generalised Riemann Hypothesis. The truly extraordinary, corrected version (formulated by Heilbronn, as detailed in the Royal Society Mathematical Tables 9 Indices and Primitive Roots by A. E. Western and J. C. P. Miller) reads as follows (in Bach and Shallit):Let NiMvJSJhRyomJkYkNiMiIiJGKCokJkYkNiMiIiNGLEYo, with NiMmJSJhRzYjIiIi squarefree, and let h be the largest integer for which a is a NiMpJSJoRyUjdGhH power. Let NiMvJiUiQUc2IyIiISomLSUoUHJvZHVjdEc2IywmIiIiRi0qJkYtRi0sJiUicUdGLUYtISIiRjFGMUYtLUYqNiMsJkYtRi0qJkYtRi0qJkYwRi1GL0YtRjFGMUYtwhere the first product is taken over primes q dividing hand the second is taken over primes q not dividing hthen (conjecture) NiMtJiUjcGlHNiMlImFHNiMlInhH is asymptotic to NiMqKCYlIkFHNiMiIiEiIiIlInhHRigtJSNsbkc2I0YpISIi as NiNmKjYjJSJ4RzciNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIlKWluZmluaXR5R0YqRipGKg== if NiMwJSJhRyIiIg== (mod 4)otherwise (conjecture) NiMtJiUjcGlHNiMlImFHNiMlInhH is asymptotic to NiMqKCYlIkFHNiMiIiJGJyUieEdGJy0lI2xuRzYjRighIiI= as NiNmKjYjJSJ4RzciNiQlKW9wZXJhdG9yRyUmYXJyb3dHNiIlKWluZmluaXR5R0YqRipGKg==, whereNiMvJiUiQUc2IyIiIiomJkYlNiMiIiFGJywmRidGJyooLSUjbXVHNiMtJSRhYnNHNiMmJSJhR0YmRictJShQcm9kdWN0RzYjKiZGJ0YnLCYlInFHRiciIiMhIiJGPUYnLUY3NiMqJkYnRicsKCokRjtGPEYnRjtGPUYnRj1GPUYnRj1GJw== where 'NiMlI211Rw==' is the Mobius function (in Maple that's 'mobius')the first product is taken over primes q dividing NiMtJSRnY2RHNiQmJSJhRzYjIiIiJSJoRw==and the second is taken over primes q dividing NiMmJSJhRzYjIiIi but not dividing h(see page 222 of Bach and Shallit for many references)Some Maple work:A := product(1 - 1/(ithprime(r)*(ithprime(r)-1)),r=1..infinity);evalf(A); # it is NO surprise that Maple cannot evaluate A:So, let's merely get an approximate value:A[20] := product(1 - 1/(ithprime(r)*(ithprime(r)-1)),r=1..20);evalf(A[20]);In the following I calculate the number of times for which 2 (then 3, 5, 27) is a primitive for primes up to 'x' (for various x's: NiUqJCIjNSIiIyokRiQiIiQqJEYkIiIl; one may take higher values depending on the speed of computer available), and the relative frequency (i.e., relative to NiMqJiUieEciIiItJSNsbkc2I0YkISIi ). Here I have written a simple Maple procedure, which might not be familiar to all my readers, but it ought to be clear what is going on (and I spell it out just once):Artin := proc(a, x) local count, r; count := 0: for r while ithprime(r) <= x do if order(a, ithprime(r)) = ithprime(r)-1 then count := count+1 fi od; print(count, count/evalf(x/ln(x))); end:Artin(2, 10^2);Thus there are 12 primes up to 100 for which 2 is a primitive root, and the '.552 ... ' is ... .Artin(2, 10^3);Artin(2, 10^4);Artin(3, 10^4);Here I take a case where NiMvJSJhRyIiIg== (mod 4) (for the 'otherwise')Artin(5, 10^4);Artin(6, 10^4);Artin(27, 10^4); # note the (expected) 'drop'...

This is a well-known, longstanding problem (one on which I spent a lot of time over the years). I first came upon it at school, in Sierpinski's wonderful 1964 collection of problems. There, on page 111, Sierpinski asks:Does there exist a composite number n which is a divisor of the numberNiMsJikiIiIsJiUibkdGJUYlISIiRiUpIiIjRiZGJQ== + ... + NiMsJiksJiUibkciIiJGJyEiIkYlRidGJ0Ynand Sierpinski goes on to (trivially) remark: It is easy to prove that if p is a prime number, thenNiMsJikiIiIsJiUicEdGJUYlISIiRiUpIiIjRiZGJQ== + ... + NiMsJiksJiUicEciIiJGJyEiIkYlRidGJ0Ynis divisible by p. [and that is all Sierpinski had to say about it.]The point of Sierpinski's remark is that, for prime p, one has:NiMvKSIiIiwmJSJwR0YlRiUhIiJGJQ== (mod p), NiMvKSIiIywmJSJwRyIiIkYoISIiRig= (mod p), ... , NiMvKSwmJSJwRyIiIkYnISIiRiVGJw== (mod p)and so, trivially: NiMsJikiIiIsJiUicEdGJUYlISIiRiUpIiIjRiZGJQ== + ... + NiMvLCYpLCYlInBHIiIiRighIiJGJkYoRihGKCIiIQ== (mod p) Thus the thrust of Sierpinski's question is: does NiMsJikiIiIsJiUibkdGJUYlISIiRiUpIiIjRiZGJQ== + ... + NiMvLCYpLCYlIm5HIiIiRighIiJGJkYoRihGKCIiIQ== (mod n ) ... (c)hold only for prime n?Comment. It is well known (as indeed anyone will establish who tackles this question) that a composite n satisfying (c) must have these properties:n is odd and squarefree (so NiMvJSJuRy0lKFByb2R1Y3RHNiQmJSJwRzYjJSJpRy9GKzsiIiIlInJH, for distinct odd primes NiQmJSJwRzYjIiIiJkYkNiMiIiM=, ... , NiMmJSJwRzYjJSJyRw== , NiMyIiIiJSJyRw== )NiMqJiYlInBHNiMlImlHIiIiLCZGJEYoRighIiJGKA== divides NiMvLCYmJSJuRzYjJSJpRyIiIkYpISIiLCYqJkYmRikmJSJwR0YnRipGKUYpRio= for every NiQlImlHL0YjIiIi, ... , rNow, of course, one would like to show the latter is impossible...A comment. An immediate consequence of the latter is that:NiMsJiomIiIiRiUmJSJwRzYjRiUhIiJGJSomRiVGJSZGJzYjIiIjRilGJQ== + ... + NiMsJiomIiIiRiUmJSJwRzYjJSJyRyEiIkYlKiZGJUYlLSUoUHJvZHVjdEc2JCZGJzYjJSJpRy9GMTtGJUYpRipGKg== is an integer ... (e)One might be tempted to argue that is impossible... , but one should be aware that it will not be easy. One cannot argue it is impossible merely for distinct primes, for one should be aware that:NiMvLCoqJiIiIkYmIiIjISIiRiYqJkYmRiYiIiRGKEYmKiZGJkYmIiImRihGJiomRiZGJiIjSUYoRihGJg== So, is (e) impossible for distinct odd primes?Recommended reading. The January 1996 American Mathematical Monthly article by Borwein (D., J. M., and R.) and Girgensohn, R., who attribute the above question to G. Guiga (1950), where they list ten associated Open Problems.

This proof relies on the simple, but powerful observation that for prime p, and integer NiMwJSJhRyIiIQ== (mod p), one has NiUtJSIuRzYkJSJhRyIiIi1GJDYkRiYiIiMtRiQ2JEYmIiIk, ... NiYtJSIuRzYkJSJhRywmJSJwRyIiIiIiIyEiIi8tRiQ2JEYmLCZGKEYpRilGK0YpRioiIiQ=, ... , NiQsJiUicEciIiIiIiMhIiIsJkYkRiVGJUYn (mod p) in some orderA small prime numerical demonstration:restart;p := 23;a := 12;seq(a*k mod p, k=1..p-1);sort([seq(a*k mod p, k=1..p-1)]);Then NiMvKiYpJSJhRywmJSJwRyIiIkYpISIiRiktJSpmYWN0b3JpYWxHNiNGJ0YpRis= (mod p), from which it follows that NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p).

Suppose NiMvLCYqJCUieEciIiMiIiJGKEYoIiIh (mod p) for some prime p with NiMvJSJwRyIiJA== (mod 4). Then from NiMvKiQlInhHIiIjLCQiIiIhIiI= (mod p) one has NiMvKSokJSJ4RyIiIyomLCYlInBHIiIiRishIiJGK0YnRiwpLCRGK0YsRig= (mod p), giving NiMvKSUieEcsJiUicEciIiJGKCEiIiwkRihGKQ== (mod p) ... (i) But clearly NiMwJSJ4RyIiIQ== (mod p), and so by Fermat's little theorem NiMvKSUieEcsJiUicEciIiJGKCEiIkYo (mod p) ... (ii) and (i) and (ii) are incompatible for odd p, since they imply NiMvIiIjIiIh (mod p).

If one is in a hurry then this proof allows one to sidestep having to establish all the side work necessary to a proof of the full Euler-Fermat theorem; one merely has to make two applications of the standard Fermat little theorem. We have: NiMvKSUiYUcsJiUicEciIiJGKCEiIkYo (mod p) and thus NiMvKSklImFHLCYlInBHIiIiRikhIiIsJiUicUdGKUYpRiopRilGKw== (mod p) giving NiMvKSUiYUcqJiwmJSJwRyIiIkYpISIiRiksJiUicUdGKUYpRipGKUYp (mod p) Similarly NiMvKSUiYUcqJiwmJSJwRyIiIkYpISIiRiksJiUicUdGKUYpRipGKUYp (mod q) and thus NiMvKSUiYUcqJiwmJSJwRyIiIkYpISIiRiksJiUicUdGKUYpRipGKUYp (mod pq)since NiMvLSUkZ2NkRzYkJSJwRyUicUciIiI=. [End of proof.]As one quickly learns in RSA public-key cryptography, the 2-prime version of Fermat's 'little' theorem plays a central role.

Bach, Eric and Shallit, Jeffrey. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. MIT Press, 1996.Borwein, D., J. M., and P.B., and Girgensohn, R. Giuga's Conjecture on Primality, American Mathematical Monthly 103 (1996) 40-50Cosgrave, John B., Many Maple worksheets - a number, but not all, in html format - are available at my web site; simply follow the obvious links at: http://www.spd.dcu.ie/johnbcosDickson, Leonard E. D., History of the Theory of Numbers, Volume 1 (First published in 1919). Chelsea Publishing Company (1971). Edwards, Harold M., Fermat's Last Theorem (A Genetic Introduction to Algebraic Number Theory). Springer-Verlag (1977).Gauss, Carl Friedrich. Disquisitiones Arithmeticae (originally published in 1801). Translated by Arthur A. Clarke (completed 1965). Yale University Press (1966). Reprinted by Springer-Verlag (1986).Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat 1601-1665. Princeton University Press (1973)Sierpinski, Waslaw. A Selection of Problems in the Theory of Numbers. Pergamon Press (PWN-Polish Scientific Publishers) (1964)Weil, Andr\351. Number Theory (An approach through history. From Hammurapi to Legendre.) Birkha\374ser (1984).Williams, Hugh C,. Edouard Lucas and Primality Testing. Canadian Mathematical Society Series Of Monographs And Advanced Texts, Volume 22. John Wiley & Sons, Inc. (1998)

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