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2 For r = 2, the proposer provided an elementary proof that a 2-coloring of the integer lattice contains n points of the same color whose centroid also has that color. (Note that any n points are the vertices of a simple polygon, not necessarily convex.) Given a 2-coloring, let a1,..., an be the vectors of n red points, and let C = (1/n)Eai be their centroid. If C is blue, let bj = (n+1)aj-eai for 1 < i < n. Then a1 is the centroid of the set obtained from {a1, a2,.. -, an) by replacing a1 by bj. If any bj is red, we have the desired red set. Otherwise, {bj} is a blue set with blue centroid C, by straightforward computation. Note that this proof yields the desired monochromatic figure within a very small grid. Solved also by A. Bialostocki, R. J. Chapman (Great Britain), R. High, L. Piepmeyer (Germany), and B. Reznick. Some Strange 3-adic Identities 6625 [1990, Proposed by Nicholas Strauss, Pontificia Universidade Cat6lica do Rio de Janeiro, Brasil, and Jeffrey Shallit, Dartmouth College. If k is a positive integer, let 3v(k) be the highest power of 3 dividing k. Put r(n) = for positive integers n. Prove that (i) v(r(n)) > 2v(n), (ii) v(r(n)) = v((2n)) + 2v(n). f( i() i=o Solution by Don Zagier, University of Maryland, College Park, and Max-Planck- Institut fur Mathematik, Bonn, Germany. The assertion of the problem may be stated in the form: u { E k)= (n2 (2n)) for all n > 1; (1) here, and throughouthis solution, vu() denotes the 3-adic valuation. We give a simple proof of (1) and of various other 3-adic identities related to it. If we set n-1 2k) k =0 k f (n) (n>1), (2) n2(~ then (1) says that f(n) is a 3-adic unit for all n c N. In fact, a calculation of the first few values suggests that in fact f (n) -1 (mod 3) Vn (3) and a more extensive calculation suggests the more precise congruences n -m (mod 3i) f(n) f(m) (mod 3j+1). (4) This says that the function f: N (li c 03 extends to a 3-adic continuous map Z3. The range studied (n < 2200) permits one to check these congruences for j < 7 (since 37 < 2200) and hence to interpolate f(n) with accuracy 66 PROBLEMS AND SOLUTIONS [January

3 0(38). The interpolated values found in this way for negative integers and half-integers are equal, to this accuracy, to simple rational numbers, suggesting the further identities 7 f(-1) = -1, f(-2) = -4, f(-3) = -4,... (5) f(1) - f(3)f( 5) 196 (6) We now state a result which includes all of these experimental observations. Theorem. The function f extends to a 3-adic analytic function from Z3 to Z3. Its values at negative integers and half-integers are rational numbers, given by (2n - 1)! n-i!2 f(-n) - n!2 k=o (k - 1)! (n > 1), (7) f(~ 2n2 k=o k) (n >0) 1 (8) ( 2 ) (2n + )2n() k As a corollary, we get the identities analogous to (1) n- k!2 n!2 (lkf o(2k + 1)! =v((2n -1)! (n>l) (9 v( 2 ( k ) (2n + ) (n > 0). (10) Proof: Equation (2) implies that f(n) satisfies the recursion relation (2n + 1)(2n + 2)f(n + 1) = 1 + n2f(n) ( 1) for n Ec N. If f has an extension to a 3-adic continuous function from Z3 to Z3, then this functional equation must hold for all n E Z3. Since the left-hand side vanishes at n = -1 and n = -1/2, we must have f(-1) = -1 and f(-1/2) = -4; the further values in (7) and (8) then follow by induction on n using the functional equation (11). Thus we need only prove the first statement of the theorem. Set g(n) = 2nf(n); we show firs that g extends to a 3-adic analytic function of n, and then that g(x) is divisible by x. For g the recursion (11) becomes Define rational numbers ao = 1, a1 = 2(2n + 1)g(n + 1) = 2 + ng(n). (12) - 1/2,... by requiring that g(n)= a( n ) (13) for n = 1, 2,... (note that the sum is finite for each n). If we show that v(ar) o as r oo, then (13) will converge 3-adically for all n E /3 and give the desired continuation. Substituting (13) into (12) gives 2 +o (r 1)ar n r: + 2(2r + 1) n + 4( r + 1) (n 1) ar r=or=rr 1 Comparing coefficients of (n) in this gives 2(2r + 1)ar = -3rari for r > 1, 1992] PROBLEMS AND SOLUTIONS 67

4 whence ( )r!2 (23r! (r > O). (14) ar =(2r + 1)! The 3-adic valuation of this does indeed tend to infinity with r (since v(3r/(2r + 1)!) > 0 and v(r!) -* oo), so (13) gives the analytic continuation of g. Lemma. The series Er =0(3 rr!2/(2r + 1)! ) converges 3-adically to 0. We will prove the lemma in a moment. Assuming it, we find n-1 r r! g(n ) -L 3)r (2r + 1)! (n -)(n -)..(n-r n-1 3rr!2 1 r=o (2r + 1)! 2 n-1 r + r=2 (3)r (2r + 1)! [(n - l)(n -2) (n - r) - (-1)rr!]. (15) By the lemma, the first term in (15) has valuation (n-i 3rr!2 (0 3rr!2 n -2 iro(2r + 1> 1!,2 > v(n) +1 (n >4) tr=o 2r+l)! r=n (2r + )!3 since v((3rr!2)/(2r + 1)!) > 2v(r!) > 2(r - 2)/3 for all r. Also, (n - 1)(n - 2).. (n - r) _ (-_1)r r!1 is divisible by n and (-3)rr!/(2r + 1)! is divisible by 3 for all r > 2, so (15) gives g(n) = - 'n (mod 3v(n)+ 1) whence f(n) = g(n)/2n is 3-integral and congruent to - 1 modulo 3. Thus the theorem is proved. Proof of Lemma: We have the power series identity E (2 + I)!Xr= E (ftr( - t)r dt)xr (beta integral) r=o (2r + 1)! t)o dt Jo 1+xt + xt x + x2-4x - log Ix2-4x 2 - x - VX2-4x 1 (2-x? lx2-4x)/4 3A/x2-4x log(2 -x- x2 4x )3/4 log 1 2 -x(3 -x)2 +(3 -x)(1-) X)2X2-4x 35x2-4x lg2 -x(3 -x)2 -(3 -x)(1 -x) x2-4x = ~~~~~~~~_ Xv VX~~ 2 X2 1 xn(x - 4) (3x)2n (1 -x)2n E ~~~~~~~~2n+1 3 n=o 2n + 1 [2 - x(3 - x)2] 68 PROBLEMS AND SOLUTIONS [January

5 in Q[[x]]. Both sides converge 3-adically if v(x) > 0, and the right-hand side vanishes for x = 3. This completes the proof of the lemma. Finally, we remark that the computer calculations to n = 2200 suggested the further congruence n m 0 (mod3') * f(n) -f(m) (mod32i+'), analogous to (4). If true, this says that the derivative of f at 0 vanishes. From what we have done we find that the Taylor series of f around the origin is given by rr!2 [n f \/ ln 2n r=o (2r + 1)! 2 r with = A + Bn + Cn2 + * 1 3rr! ra7 2 1 (2r + 1)! 1+ r rr!2 2 r=2 (2r + 1)! 2 r etc. (o-2= second elementary symmetric function). The assertion that f'(0) vanishes is thus equivalent to the following statement, which is similar to but more complicated than our lemma above: Conjecture. The series Er=0((3rr!2)/(2r + 1)! )o-2(1, 1/2,..., 1/r) converges 3-adically to 0. Another interesting problem would be to evaluate in closed form the 3-adic number A. To thirty 3-adic digits, A equals Part (i) was solved also by Derek Hacon and Nicholas Strauss. Part (ii) was solved also by Jean-Paul Allouche and Jeffrey Shallit. A Convergent Sequence E 3388 [1990, Proposed -by Matthew Cook (student), University of Illinois, Urbana, IL, Walther Janous, Ursulinengymnasium, Innsbruck, Austria, and Marcin E. Kuczma, University of Warsaw, Warsaw, Poland. Let x1 and x2 be arbitrary positive numbers. Suppose we define a sequence {xn}ln= by putting xn,2 = 2/(xn+1 + x) for n = 1, 2,3,... Prove that the sequence converges. Solution by David Borwein, University of Western Ontario, London, Ontario, Canada. We first prove that the sequence is bounded. If both xn and xn,1 are between a-1 and a, then a' < (xn + xn_)/2 < a, so Xn+1 is between the same bounds. Now let l = lim inf xn and L = lim sup xn. Since L is finite, for any E > 0 there is an integer no such that xn < L + E for n > no. Hence xn+2 = 2/(xn+1 + xd) > 1/(L + e) for n > no. It follows that 1 > 1/L > 0. Similarly, Xn > l - 8 > 0 for n > n1 implies xn+2 < 17(1-8) for n > n1, whence L < 1/1. Therefore 1= 1/L. 1992] PROBLEMS AND SOLUTIONS 69

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