Fermat's Last Theorem Implies Euclid's Infinitude of Primes
aa r X i v : . [ m a t h . N T ] S e p Fermat’s Last Theorem ImpliesEuclid’s Infinitude of Primes
Christian Elsholtz
Abstract.
We show that Fermat’s last theorem and a combinatorial theorem of Schur onmonochromatic solutions of a + b = c implies that there exist infinitely many primes. In par-ticular, for small exponents such as n = 3 or this gives a new proof of Euclid’s theorem,as in this case Fermat’s last theorem has a proof that does not use the infinitude of primes.Similarly, we discuss implications of Roth’s theorem on arithmetic progressions, Hindman’stheorem, and infinite Ramsey theory towards Euclid’s theorem. As a consequence we see thatEuclid’s Theorem is a necessary condition for many interesting (seemingly unrelated) resultsin mathematics.
1. INTRODUCTION.
Imagine that the set of positive integers has only finitely manyprimes. We will investigate consequences, and to become more creative with this, weimagine we live in an entirely different world, namely in a “world with only finitelymany primes.” If you are a number theorist, then you will realize that a major partof analytic number theory just vanishes. One of the implications of this article is thatalgebraic number theorists and combinatorialists would live in a very different world,too. The reason is that “Fermat’s last theorem” even in the first interesting case withexponent would be wrong, that major parts of the modern subject of additive combi-natorics would disappear, and that even basic results of infinite Ramsey theory wouldnot exist. If you wonder why this is the case, we invite you to a journey of unexpecteddiscoveries in the fictional “world with only finitely many primes”!There are many proofs of Euclid’s theorem stating that there exist infinitely manyprimes. There is a very thorough bibliographic collection of 70 pages on a multitudeof proofs of Euclid’s theorem, due to Meˇstrovi´c [ ]. Other collections are given byRibenboim [ ] and a very recent one by Granville [ ]. For some recent proofs, see[
27, 33 ].Many of these proofs make use of an infinite sequence with mutually coprime inte-gers, such as F n = 2 n + 1 (Goldbach, in a letter to Euler 1730), or primitive divisorsof certain recursive sequences (see, e.g., [ ]). Furstenberg [ ] made use of a suitablydefined topology to prove Euclid’s theorem. A number of proofs have used the expo-nents of a prime factorization; see, for example, [
10, 11, 22 ]. Even more recently, twoproofs [
1, 18 ] made use of van der Waerden’s theorem applied to the patterns of expo-nents . Alpoge [ ] introduced van der Waerden’s theorem to this subject, and Granville[ ] combined Alpoge’s idea with a theorem of Fermat, namely that there are no foursquares in arithmetic progression.Inspired by this new type of proof, we investigate which type of purely combina-torial results can be combined with some kind of arithmetic result to give new proofsthat there exist infinitely many primes. In this way we link Euclid’s theorem to somevery beautiful and significant results of modern mathematics.Here is a brief outline of the article. In Section 2 we link Euclid’s theorem to Fer-mat’s last theorem, eventually proved by Wiles [ ], and to a theorem of Schur (1916),which is often considered to be the starting point of combinatorial number theory.In Section 3 the link is to a theorem of Roth (1953) on the density of integers with-out arithmetic progressions. An independent elementary proof of Euclid’s theorem is1 by-product, in Section 4. Section 5 has some discussion about varying the number-theoretic or combinatorial input. Section 6 uses a theorem of Hindman (1974) on aninfinite extension of Schur’s theorem, and Section 7 gives two proofs using infiniteRamsey theory.Roth’s theorem and its extension by Szemer´edi [ ], and quantitative versionsthereof, (e.g., due to Bourgain [ ], Gowers [ ], Green and Tao [ ]) has inspiredmany excellent mathematicians and has had tremendous impact on the relativelyyoung field of additive combinatorics.
2. FERMAT’S LAST THEOREM IMPLIES EUCLID’S THEOREM.
We firststate Schur’s theorem and then the main result of this article.
Lemma 1 (Schur’s theorem [28], 1916).
For every positive integer t , there exists aninteger s t such that if one colors each integer m ∈ [1 , s t ] using one of t distinct colors,then there is a monochromatic solution of a + b = c, where a, b, c ∈ [1 , s t ] . Theorem 1.
For n ≥ let FLT( n ) denote the statement “There are no solutions ofthe equation x n + y n = z n in positive integers x, y, z .” Then “ FLT( n ) is true” andSchur’s theorem imply that there exist infinitely many primes. Theorem 1 gives a new proof of Euclid’s theorem for those exponents for which aproof of
FLT( n ) independent of the infinitude of primes exists. This is certainly thecase for n = 3 , , , where elementary proofs exist (see [
9, 24 ]). It then trivially fol-lows for infinitely many exponents, for example for all multiples of . The applicationof Fermat’s last theorem with general n to Euclid’s theorem might possibly competefor the most indirect proof, but at present the proof with general n is not actually aproof at all, as Wiles’s proof makes use of the fact that there exist infinitely manyprimes.We briefly show that Schur’s theorem nowadays can be seen as a direct conse-quence of Ramsey’s theorem [ ] (1929). Ramsey’s theorem (see [ , Theorem 10.3.1])states that, for any number t of colors (let us call them , . . . , t ) and positive integers n , . . . , n t there exists an integer R ( n , . . . , n t ) such that if the edges of the completegraph on R ( n , . . . , n t ) vertices are colored, there exists an index i and a monochro-matic clique of size n i all of whose edges are of color i . In our application we onlyneed the case n = · · · = n t = 3 .Let χ : { , . . . , N } → { , . . . , t } be the coloring of the first N = R (3 , . . . , integers. Let us define a coloring of the edges of the complete graph with vertices { , , . . . , N } as follows: The edge ( i, j ) is given the color χ ( | i − j | ) . Ramsey’stheorem guarantees that there is a monochromatic triangle. Let us denote the vertices ofthis triangle by ( i, j, k ) , where i < j < k . Let a = j − i, b = k − j , and c = k − i .Then a, b, c all have the same color and a + b = c holds. This gives the requiredmonochromatic solution. Proof of Theorem 1.
Suppose there exist only finitely many primes p , . . . , p k (say).Every postive integer can be written as m = Q ki =1 p e i i . We write integers as an n th power times an n th power-free number. Hence, writing e i = nq i + r i with ≤ r i ≤ n − gives m = (cid:16)Q ki =1 p q i i (cid:17) n (cid:16)Q ki =1 p r i i (cid:17) = N ( m ) × R ( m ) (say). Weuse n k distinct colors, denoted by ( t , . . . , t k ) , ≤ t i ≤ n − , and we color the inte-ger m = Q ki =1 p nq i + r i i by ( r , . . . , r k ) . By Schur’s theorem there exists a monochro-matic triple ( a, b, c ) such that c = a + b and with a fixed color ( r , . . . , r k ) , cor-responding to R = Q ki =1 p r i i . Here a, b, c all contain the same factor R and we2an write a, b, c as a = N ( a ) R, b = N ( b ) R, c = N ( c ) R , with positive integers N ( a ) , N ( b ) , N ( c ) . Dividing by R gives N ( a ) + N ( b ) = N ( c ) with n th powers,which is a contradiction to FLT( n ) . It might seem that we require unique factorization, as for an integer with distinctprime factorizations the coloring is not well-defined. However, for an application ofSchur’s theorem it is perfectly fine if an integer m with hypothetical distinct primefactorizations is assigned only one of the colors. (Assigning all corresponding colorsto m would be an alternative, but then χ would not actually be a function.)It is of historic interest to note that Schur’s motivation was to study Fermat’s equa-tion modulo primes. Dickson had proved that there is no congruence obstruction to theFermat equation, and Schur [ ] gave a simple proof of this.
3. ROTH’S THEOREM IMPLIES EUCLID’S THEOREM.
The Fermat equationhas also been studied with coefficients. The case x n + y n = 2 z n in positive integershas attracted special attention, as a solution in distinct positive integers would meanthat there exist n th powers x n < z n < y n in arithmetic progression. It was conjec-tured by D´enes that for n ≥ there exist only trivial solutions with x = y = z . Thiswas proved by Darmon and Merel [ ] based on the methods of Wiles. Sierpi´nski [ ]gives elementary proofs of the cases n = 4 (Chapter 2, § ) and n = 3 (Chapter 2, § ); see also [ ]. We also give new proofs of Euclid’s theorem in these cases.The following result gives a matching combinatorial tool. Lemma 2 (Roth [26]).
Let δ > and N ≥ N ( δ ) . Every subset S ⊂ [1 , N ] of at least δN elements contains three distinct elements s , s , s ∈ S in arithmetic progression,i.e., s + s = 2 s . It should be noted that there is a purely combinatorial proof of Roth’s theorem, e.g.,in [ , pp. 46–49]. In contrast to van der Waerden’s and Schur’s theorem the abovestatement is a so-called “density version”: this result not only guarantees monochro-matic solutions in some unspecified color, but even in all those colors that occur witha positive density. Theorem 2.
For n ≥ let DM( n ) denote the statement “There are no three positive n th powers in arithmetic progression” or equivalently “There are no solutions of theequation x n + y n = 2 z n in positive integers x < z < y .” Then “ DM( n ) is true” andRoth’s theorem imply that there exist infinitely many primes.Proof. We first prove the following (possibly surprising) lemma.
Lemma 3.
Suppose there exist only finitely many primes p < · · · < p k . The set of n thpowers is a positive proportion of all integers, i.e., there exists some δ = δ ( n, k ) > such that for all N the set of n th powers in [1 , N ] is at least δN .Proof of Lemma. We prove this by dividing a lower bound approximation of the num-ber of n th powers in [1 , N ] by an upper bound approximation of all integers in [1 , N ] ,both counted by means of exponent patterns. The upper bound on the number of pos-sible exponent patterns ( e , e , . . . , e k ) follows from p e · · · p e k k ≤ N , which gives e i ≤ log N log p i . Hence (1 + log N log p ) · · · (1 + log N log p k ) is an upper bound. For the lower boundon the number of n th powers, we count those e i divisible by n and with p e i i ≤ N /k for all i = 1 , . . . , k . We see that at least ⌊ log Nnk log p ⌋ · · · ⌊ log Nnk log p k ⌋ of all inte-gers at most N are n th powers, which gives (for large N ) a positive proportion of atleast δ ≥ lower boundupper bound ≥ C ( nk ) k , for some C > .3ith this lemma we can replace Schur’s theorem by Roth’s theorem. Roth’s the-orem directly guarantees that there exists a nontrivial arithmetic progression of n thpowers, which is in contradiction to DM( n ) . (Note that in this case there is no need todivide by the factor R of the first proof.) Remark.
The results by van der Waerden (used by Alpoge and Granville) and Schuror Roth (used here) are early results of Ramsey theory. The numerical bounds on s t implied by Schur’s theorem are moderate, compared to the very quickly increasingbounds in van der Waerden’s theorem. Let s t denote the least number such that forany t -coloring, which is a map χ : { , . . . , s t } → { , . . . , t } , there exist a, b, c with a + b = c and χ ( a ) = χ ( b ) = χ ( c ) . It follows from Schur’s proof that s t ≤ ⌊ t ! e ⌋ .
4. POSITIVE DENSITY GIVES A NEW ELEMENTARY PROOF.
The obser-vation about “positive density” in Lemma 3 also leads to a short and new proof ofEuclid’s theorem:
Proof.
Lemma 3 says the number of n th powers (for any fixed n ≥ ) has positivedensity in the set of positive integers. But it is also clear that there are at most N /n positive n th powers x n ≤ N , contradicting the lower bound of δN (for some fixed δ > ) for sufficiently large N . Comparing with the bibliography [ ], the proof clos-est in spirit appears to be Chaitin’s proof [ ].We note that the main focus of this article is not about short proofs but how seem-ingly remote results can be applied.
5. DISCUSSION ON VARIANTS OF THE PROOFS ABOVE: THE FRANKL–GRAHAM–R ¨ODL THEOREM AND FOLKMAN’S THEOREM.
1. We now discuss that knowing something more on the combinatorial side, namelyknowing about the number of monochromatic solutions, helps in reducing thenumber-theoretic input considerably.On the combinatorial side, Frankl, Graham, and R¨odl [ ] proved that with t colors the number of monochromatic solutions ( a, b, c ) of the equation a + b = c with a, b, c ∈ [1 , N ] increases quadratically, i.e., there is a positive constant c t such that the number S ( t, N ) of solutions is at least c t N . (In fact, [ ]gives a direct proof for the Schur equation, but also covers much more generalcases.) As in the proof of Theorem 1, the monochromatic solutions of a + b = c correspond to solutions of x n + y n = z n in positive integers.On the number-theoretic side there are several reasons why the number ofsolutions is smaller, giving a contradiction to the assumption “there are finitelymany primes only.”A result of Faltings [ ] would give there are at most O ( N ) solutions of x n + y n = z n with x n , y n , z n ∈ [1 , N ] , being coprime in pairs. A much moreelementary approach is as follows: For odd n the left-hand side of x n + y n = z n can be factored as ( x + y ) P n − i =0 ( − i x n − − i y i . In particular, when n = 3 thisis x + y = ( x + y )( x − xy + y ) . The number of divisors of any integer z n ≤ N is clearly at most √ N . (Actually, as we assume there are at most k prime factors, this can be improved to C k (log N ) k .)Hence the number-theoretic upper bound of at most N values of z n with atmost √ N factorizations each and the combinatorial lower bound of at least c t N solutions contradict each other. 4his remark also applies in the situation of x n + y n = 2 z n , as using a resultof Varnavides [ ] one can also prove that in this situation there would be at least c t N many solutions, with x, y, z ≤ N , contradicting as before the number-theoretic upper bound.2. For the combinatorial lemma there are other alternatives. For example, a theoremof Folkman [ , p. 81] guarantees much larger monochromatic structures thanSchur’s theorem does: For every number t of colors, every coloring χ : N →{ , . . . , t } , and every s ∈ N , there exist N s,t and a , . . . , a s ∈ [1 , N s,t ] withthe property that all nontrivial subset sums P i ∈ I a i , where I ⊆ { , . . . , s } isnonempty, are monochromatic. In analogy with the proof of Theorem 1, thiswould mean, in the special case s = 3 , applied with the same coloring and afterdividing by the common factor R , that all of a ′ , a ′ , a ′ , a ′ + a ′ , a ′ + a ′ , a ′ + a ′ , a ′ + a ′ + a ′ are n th powers. Proving that this is impossible could be easierthan proving FLT( n ) , as FLT( n ) corresponds to s = 2 with fewer conditions.But we are not aware of any literature on this.
6. HINDMAN’S THEOREM IMPLIES EUCLID’S THEOREM.
Let us explic-itly write down an extreme form of the above remark on Folkman’s theorem. An ex-tention of Folkman’s theorem is Hindman’s theorem [ ]; see also [ ] and [ , p. 85]. Lemma 4.
For any integer t ≥ and any t -coloring χ : N → { , . . . , t } , there ex-ists an infinite sequence A = { a , a , . . . } such that all subset sums P i ∈ I a i overnonempty finite index sets I ⊂ N are monochromatic. Theorem 3.
Hindman’s theorem implies Euclid’s theorem.Proof.
We start as in the proof of Theorem 1. Suppose there exist only finitely manyprimes p , . . . , p k (say). Every integer can be written as m = Q ki =1 p e i i , e i = nq i + r i with ≤ r i ≤ n − . That is, m = (cid:16)Q ki =1 p q i i (cid:17) n (cid:16)Q ki =1 p r i i (cid:17) = N ( m ) × R ( m ) (say). We color the integer m = Q ki =1 p nq i + r i i by ( r , . . . , r k ) . By Hindman’s theoremthere exists an infinite set such that all nonempty finite subset sums are monochromaticwith a fixed color ( r , . . . , r k ) , corresponding to R = Q ki =1 p r i i . Dividing by R givesan infinite set such that all finite subset sums are n th powers.This would in particular correspond to some fixed x n and infinitely many pairs ( y ni , z ni ) of n th powers such that x n + y ni = z ni holds. This is clearly impossible, asthe difference between consecutive n th powers z n − ( z − n ≥ z n − increases when n ≥ is fixed and z increases. Remark.
The proof of Hindman’s theorem is not trivial, but it is certainly much moreaccessible than
FLT( n ) for general n . Moreover, the proof of Hindman’s theoremdoes not make use of Euclid’s theorem, in contrast to Wiles’s proof of FLT .
7. INFINITE RAMSEY THEORY IMPLIES EUCLID’S THEOREM.
Theabove proof does not need the full strength of Hindman’s theorem, as it essentiallyonly uses sums of two elements. Hence it is possible to reduce the combinatorial inputaccordingly, which we discuss below.
Lemma 5 (The Infinite Ramsey Theorem
IRT , see e.g., [8, Theorem 9.1.2]).
Let X be some infinite set and color all subsets of X of size w with t different colors. Thenthere exists some infinite subset M ⊂ X such that the subsets of M of size w all havethe same color.
5n plain words, the case w = 2 of Lemma 5 says that a finite coloring of the com-plete graph K ∞ guarantees a complete monochromatic K ∞ as a subgraph. Theorem 4.
The infinite Ramsey theorem
IRT implies Euclid’s theorem.
We leave the proof of Theorem 4 as an exercise to the reader, and only remark it isa variant of the Theorem 3 and our final Theorem 5.It turns out that one does not actually need an infinite complete monochromaticgraph, but only a monochromatic complete bipartite graph K , ∞ , where one set of thevertices consists of two elements and the other one is infinite (say countable).We give a complete proof of this and the application to Euclid’s theorem below. Toprove the existence of this infinite substructure is quite simple. Lemma 6 (The K , ∞ Lemma).
Let X be some infinite set and color all pairs of twodistinct elements of X with t different colors. Then there exist a set V = { v , v } ⊂ X and an infinite set W = { w , w , . . . } ⊂ X \ V such that all edges ( v i , w j ) , with i ∈ { , } and j ∈ N , have the same color. For ease of notation we assume that X is countable. Proof.
One can construct the required sets step by step.Choose any set A = { a , a , . . . , a t +1 } ⊂ X of t + 1 distinct elements as vertices.Let v = a . There are infinitely many adjacent edges ( v , x j ) . Hence one of the t colors, say color c , occurs infinitely often. Let X = { x ,j : j ∈ N } ⊂ X be theset of those elements such that ( v , x ,j ) are these infinitely many edges of color c .Now study the color of all ( a i , x ,j ) as follows. There exists one color c (say) thatoccurs infinitely often among the infinitely many edges ( a , x ,j ) . Let X = { x ,j : j ∈ N } ⊂ X be those elements such that ( a , x ,j ) are of color c . If c = c wehave found the required substructure with V = { a , a } and W = X . We thereforeassume that c = c . We iterate the step above and come to infinite subsets X t +1 ⊂ X t ⊂ · · · ⊂ X ⊂ X ⊂ X ⊂ X such that for fixed i all edges ( a i , x i,j ) , j ∈ N , areof color c i (say). As there are t distinct colors only, there must be two distinct indices i , i ∈ { , . . . , t + 1 } such that c i = c i . With i < i without loss of generalityand V = { a i , a i } , W = X i and the lemma is proved.An alternative is to color the elements x ∈ X \ A with the vector color ( c , . . . , c t +1 ) if the color of the edge ( a i , x ) is c i , i = 1 , . . . , t + 1 . As there is only a finite numberof vector colors, namely t t +1 , there is an infinite number of x ∈ X \ A with the samevector color, which defines the set W . As before, there are two indices i = i suchthat c i = c i . Hence V = { a i , a i } and W are the sets required. Theorem 5.
The K , ∞ lemma implies Euclid’s theorem.Proof. Let n ≥ , and assume that p , . . . , p k is the list of all primes. We colorthe integers by the same rule as before: m = Q ki =1 p nq i + r i i is colored by χ ( m ) =( r , . . . , r k ) . Based on this coloring we define an infinite graph on the positive inte-gers. The edges ( m i , m j ) receive the color χ ( m i + m j ) .We apply the K , ∞ lemma to this graph: there exists a complete bipartite graph withparts V = { v , v } and an infinite set W such that all edges ( v i , w j ) , with i ∈ { , } and j ∈ N , have the same color ( r , . . . , r k ) .We multiply all integers in N by the constant P = Q ki =1 p n − r i i . All pairwise sums P v i + P w j = P ( v i + w j ) are an n th power z ni,j (say). Note that z n ,j − z n ,j = P ( v − v ) is a constant, and is also the distance between infinitely many distinctpairs of n th powers, for the infinitely many values j . This is impossible, as the gapbetween consecutive n th powers increases (see above).6ith Hindman’s theorem we made use of a quite advanced combinatorial result, andthe number-theoretic part became correspondingly quite simple. We then reduced thedepth of the combinatorial lemma until we reached the K , ∞ lemma. On the number-theoretic side, we eventually used the elementary fact that the gaps between consecu-tive n th powers increase and simple arithmetic such as P ( m i + m j ) = P m i + P m j .
8. CONCLUSION.
As our journey through a fictional world comes to an end, let usbriefly reflect: a common theme in all variants discussed is that the existence of onlyfinitely many primes would guarantee patterns for the set of n th powers that cannotactually exist, sometimes for deep reasons, sometimes for obvious ones, depending onthe strength of the pattern. Summarizing the results we find: Corollary 1.
In the “world with only finitely many primes” the following hold: If Schur’s theorem holds, then
FLT( n ) is wrong for all n ≥ .If FLT( n ) holds for some n ≥ , then Schur’s theorem does not hold. If Roth’s theorem holds, then
DM( n ) is wrong for all n ≥ .If DM( n ) holds for some n ≥ , then Roth’s theorem does not hold. The set of n th powers has positive density (giving an immediate contradiction). Hindman’s theorem does not hold. The infinite Ramsey theorem (IRT) does not hold. The K , ∞ lemma does not hold. In other words, Euclid’s theorem is logically connected with many interesting andseemingly unrelated results in mathematics.Having seen all these variants and extensions, the original version, i.e., the combi-nation of Schur’s theorem and the Fermat–Wiles theorem is the one that looks mostintriguing to this author. And Fermat’s last theorem may be the one that many of uswould miss most in the fictional “world with only finitely many primes”!
ACKNOWLEDGMENTS.
The author would like to thank the referees, the editor, R. Dietmann, J. Erde,I. Leader, R. Meˇstrovi´c, J.-C. Schlage-Puchta and A. Wiles for useful comments on the manuscript. The authorwas partially supported by the Austrian Science Fund (FWF): W1230 and I 4945-N.
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