Squares in arithmetic progressions and infinitely many primes
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Squares in arithmetic progressions andinfinitely many primes
Andrew Granville
Abstract.
We give a new proof that there are infinitely many primes, relying on van der Waer-den’s theorem for coloring the integers, and Fermat’s theorem that there cannot be four squaresin an arithmetic progression. We go on to discuss where else these ideas have come togetherin the past.
1. INFINITELY MANY PRIMES
Levent Alpoge recently gave a rather differentproof [ ] that there are infinitely many primes. His starting point was the famous resultof van der Waerden (see, e.g., [ ]): van der Waerden’s Theorem. Fix integers m ≥ and ℓ ≥ . If every positive integeris assigned one of m colors, in any way at all, then there is an ℓ -term arithmeticprogression of integers which each have the same color. Using a clever coloring in van der Waerden’s theorem, and some elementary numbertheory, Alpoge deduced that there are infinitely many primes. We proceed from van derWaerden’s theorem a little differently, employing a famous result of Fermat (see, e.g.,[ ]): Fermat’s Theorem.
There are no four-term arithmetic progressions of distinct integersquares.
From these two results we deduce the following:
Theorem 1.
There are infinitely many primes.Proof.
If there are only finitely many primes p , . . . , p k , then every integer n can bewritten as p e · · · p e k k for some integers e , e , . . . , e k ≥ . We can write each of theseexponents e j as e j = 2 q j + r j , where r j is the “remainder” when dividing e j by 2,and equals 0 or 1. Therefore if we let R = p r · · · p r k k then R is a squarefree integer that divides n , and n/R is the square of an integer . (In fact, n/R = Q where Q = p q · · · p q k k .)We will use k colors to color the integers: Integer n is colored by the vector ( r , . . . , r k ) . By van der Waerden’s theorem there are four integers in arithmetic pro-gression A, A + D, A + 2
D, A + 3 D, with D ≥ , January 2014]
PRIMES AND PROGRESSIONS athematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:24 a.m. MonthlyNoteUsingTheirStyleFile.v2.tex page 2 which all have the same color ( r , . . . , r k ) . Now R = p r · · · p r k k divides each of thesenumbers, so also divides D = ( A + D ) − A . Letting a = A/R and d = D/R , wesee that a, a + d, a + 2 d, a + 3 d are four squares in arithmetic progression,contradicting Fermat’s theorem.These ideas have come together before to make a rather different, not-too-obviousdeduction:
2. THE NUMBER OF SQUARES IN A LONG ARITHMETIC PROGRESSION
Let Q ( N ) denote the maximum number of squares that there can be in an arithmeticprogression of length N . A slight refinement of the Erd˝os–Rudin conjecture states thatthe maximum number is attained by the arithmetic progression { n + 1 : 0 ≤ n ≤ N − } which contains q N squares, plus or minus one. From Fermat’s theorem one easilysees that Q ( N ) ≤ N + 34 , but it is difficult to see how to improve the bound to, say, Q ( N ) ≤ δN + b for someconstant δ < .It was this problem that inspired one of the most influential results [ ] in combina-torics and analysis (see, e.g., [ ]): Szemer´edi’s Theorem.
Fix δ > and integer ℓ ≥ . If N is sufficiently large (de-pending on δ and ℓ ) then any subset A of { , , . . . , N } with ≥ δN elements, mustcontain an ℓ -term arithmetic progression. van der Waerden’s theorem is a consequence of Szemer´edi’s theorem, because if welet δ = 1 /m and we color the integers in { , , . . . , N } with m colors, then at leastone of the colors is used for at least N/m integers. We apply Szemer´edi’s theorem tothis subset A of { , , . . . , N } , to obtain an ℓ -term arithmetic progression of integerswhich each have the same color.In [ ], Szemer´edi applied his result to the question of squares in arithmetic progres-sions: Theorem 2 (Szemer´edi).
For any constant δ > , if N is sufficiently large, then Q ( N ) < δN .Proof. Suppose that there are at least δN squares in the arithmetic progression { r + ns : n = 1 , , . . . , N } with s ≥ ; that is, there exists a subset A of { , , . . . , N } with at least δN elements for which r + ns is a square, whenever a ∈ A. Szemer´edi’s theorem with ℓ = 4 then implies that A contains a four-term arithmeticprogression, say u + jv for j = 0 , , , . For these values of n , we have r + ns = a + jd , where a = r + us and d = vs > . That is, we have shown that a, a + d, a + 2 d, a + 3 d are four squares in arithmetic progression,contradicting Fermat’s theorem.2 c (cid:13) THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121 athematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:24 a.m. MonthlyNoteUsingTheirStyleFile.v2.tex page 3
3. MORE HEAVY MACHINERY
One day over lunch, in late 1989, Bombierishowed me a completely different proof of Theorem 2, this time relying on one of themost influential results in algebraic and arithmetic geometry, Faltings’ theorem [ ].Faltings’ theorem is not easy to state, requiring a general understanding of an algebraiccurve and its genus. The basic idea is that an equation in two variables with rationalcoefficients has only finitely many rational solutions (that is, solutions in which thetwo variables are rational numbers), unless the equation “boils down to” an equationof degree 1, 2 or 3. To be precise about “boiling down” involves the concept of genus,which is too complicated to explain here (see, e.g., [ ]). Here we only need a simpleconsequence of Faltings’ theorem. Corollary to Faltings’ Theorem.
Let b , b , . . . , b k be distinct integers with k ≥ .Then there are only finitely many rational numbers x for which ( x + b )( x + b ) · · · ( x + b k ) is the square of a rational number . Another proof of Theorem 2.
Fix an integer
M > /δ . Let B ( M ) be the total num-ber of rational numbers x and integer 6-tuples b = 0 < b < . . . < b ≤ M − forwhich ( x + b )( x + b ) · · · ( x + b ) is the square of a rational number. Faltings’ the-orem implies that B ( M ) is some finite number, as there are only finitely many choicesfor the b j . We let N be any integer ≥ M ( B ( M ) + 5) .The interval [0 , N − is covered by the sub-intervals I j for j = 0 , , , . . . , k − , where I j denotes the interval [ jM, ( j + 1) M ) , and kM is the smallest multiple of M that is greater than N .Let N := { n : 0 ≤ n ≤ N − and a + nd is a square } , where the arithmeticprogression is chosen so that |N | = Q ( N ) . Let N j = { n ∈ N : n ∈ I j } for eachinteger j . Let J be the set of integers j for which N j has six or more elements.Now if n < n < . . . < n all belong to N j , write x = a/d + n and b i = n i − n for i = 1 , . . . , , so that b = 0 < b < . . . < b ≤ M − and each x + b i = a/d + n i = ( a + n i d ) /d , which implies that ( x + b )( x + b ) · · · ( x + b ) = ( a + n d )( a + n d ) · · · ( a + n d ) d is the square of a rational number.This gives rise to one of the B ( M ) solutions counted above, and all the solutionscreated in this way are distinct (since given x, d, b , . . . , b we have each a + n j d = d ( x + b j ) ). Therefore the set N j gives rise to (cid:0) |N j | (cid:1) such solutions, and so in total wehave X j ∈ J |N j | ! ≤ B ( M ) . It is easy to verify that r ≤ (cid:0) r (cid:1) for all integers r ≥ , and so Q ( N ) = |N | = k − X j =0 |N j | ≤ k − X j =0 X j ∈ J |N j | ! ≤ k + B ( M ) , January 2014]
PRIMES AND PROGRESSIONS athematical Assoc. of America American Mathematical Monthly 121:1 August 24, 2017 12:24 a.m. MonthlyNoteUsingTheirStyleFile.v2.tex page 4 as |N j | ≤ if j J . Finally, as k ≤ N/M + 1 we have Q ( N ) ≤ k + B ( M ) ≤ NM + ( B ( M ) + 5) ≤ NM < δN, as desired.Bombieri [ ] went on, together with Granville and Pintz, to combine these twoproofs (along with much more arithmetic geometry machinery), to prove that Q ( N ) < N c for any c > , for sufficiently large N . Bombieri and Zannier [ ] improved this to c > with a rather simpler proof. The conjecture that Q ( N ) behaves more like aconstant times N / remains open. REFERENCES
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Amer. Math. Monthly (2015), 784–785.2. E. Bombieri, A. Granville, and J. Pintz, Squares in arithmetic progressions,
Duke Math. J. (1992),369–385.3. E. Bombieri and W. Gubler, Heights inDiophantine Geometry, New Mathematical Monographs, Vol. 4,Cambridge Univ. Press, Cambridge 2006.4. E. Bombieri and U. Zannier, A note on squares in arithmetic progressions. II, Atti Accad. Naz. Lincei Cl.Sci. Fis. Mat. Natur. Rend. Lincei (2002), 69–75.5. W.T. Gowers, A new proof of Szemer´edi’s theorem, Geom. Funct. Anal. (2001), 465–588.6. J.H. Silverman, Thearithmeticofellipticcurves, Springer Verlag, New York, 1986.7. E. Szemer´edi, The number of squares in an arithmetic progression, Studia Sci. Math. Hungar. (1974),417.8. E. Szemer´edi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. (1975), 199–245.9. T. Tao and Van Vu, AdditiveCombinatorics, Cambridge Univ. Press, Cambridge, 2006. ANDREW GRANVILLE
D´epartement de math´ematiques et de statistique, Universit´e de Montr´eal, CP 6128succ. Centre-Ville, Montr´eal, QC H3C 3J7, [email protected] and
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United [email protected] (cid:13)