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ab+bc+ca=0, u^2+3v^2 integers

Source: French MO 2001

April 9, 2021
number theory3D geometryalgebracomplex numbers

Problem Statement

A trio is any triple (a,b,c)(a,b,c) of nonzero real numbers satisfying ab+bc+ca=0ab+bc+ca=0. A triple is said to be reduced if a+b+c=1a+b+c=1.
[hide=Part 1] We denote by CC the set of points (x,y,z)(x,y,z) in coordinate space for which (x,y,z)(x,y,z) is a trio, and by Γ\Gamma the set of those for which (x,y,z)(x,y,z) is a reduced trio. Let OO be the origin and PP be the plane given by x+y+z=1x+y+z=1.
(a) Does there exist a trio (a,b,c)(a,b,c) such that a+b+c=0a+b+c=0? (b) Prove that CC is a union of lines passing through OO, with OO excluded. (c) Prove that Γ\Gamma is the intersection of a plane and a sphere with center OO. Describe Γ\Gamma geometrically. (d) Describe CC geometrically and sketch it. (e) Let LL be a fixed point in Γ\Gamma. If L’L’ and L′′L'' are arbitrary points on Γ\Gamma, prove that the volume VV of the tetrahedron OLL’L′′OLL’L'' is maximal when the lines OL,OL’,OL′′OL,OL’,OL'' are orthogonal, and express the coordinates of L’L’ and L′′L'' in terms of those of LL. (f) Prove that the product abcabc attains its maximum and minimum values on Γ\Gamma, and find the points at which those are attained.
[hide=Part 2] A trio (a,b,c)(a,b,c) is called rational if a,b,ca,b,c are rational, and integer if a,b,ca,b,c are integers. We say that an integer trio (a,b,c)(a,b,c) is primitive if the greatest common divisor of a,b,ca,b,c is 11.
(a) Describe the set H1H_1 of points (x,y,1)(x,y,1) such that (a,b,1)(a,b,1) is a trio. Show that the point Ω1(−1,−1,1)\Omega_1(-1,-1,1) is the center of symmetry of H1H_1. Find all points of H1H_1 with integer coordinates. (b) For each nonzero integer hh, denote by ZhZ_h the set of integer trios (a,b,c)(a,b,c) with c=hc=h. Determine ZhZ_h for h=1h=1 and h=2h=2 (c) Prove that ZhZ_h is a finite set and find the number N(h)N(h) of its elements in terms of the number of divisors of h2h^2 in Z\mathbb Z. Prove that 44 divides N(h)−2N(h)-2. (d) For every positive integer hh, denote by N’(h)N’(h) the number of integer trios (a,b,c)(a,b,c) such that at least one of a,b,ca,b,c is equal to hh. Express N’(h)N’(h) in terms of N(h)N(h) depending on the parity of hh. (e) Prove that every integer trio (a,b,c)(a,b,c) can be assigned a triple of integers (r,s,t)(r,s,t) such that rr and ss are coprime, ss is nonnegative, and a=r(r+s)t,b=s(r+s)t,c=−rst.a=r(r+s)t,\qquad b=s(r+s)t,\qquad c=-rst. State and verify the converse. For which trios (a,b,c)(a,b,c) is the triple (r,s,t)(r,s,t) not unique? (f) Determine all triples (r,s,t)(r,s,t) that are assigned to some primitive trios. Deduce that if (a,b,c)(a,b,c) is a primitive trio, then ∣abc∣|abc|, ∣a+b∣|a+b|, ∣b+c∣|b+c| and ∣c+a∣|c+a| are perfect squares. (g) For each positive integer hh, denote by P(h)P(h) the number of primitive trios (a,b,c)(a,b,c) with c=hc=h. Prove that P(h)P(h) is a power of 22. For which hh is P(h)=N(h)P(h)=N(h)? Give a sequence of integers (hn)(h_n) for which the sequence P(hn)N(hn)\frac{P(h_n)}{N(h_n)} converges to zero. (h) Let (a,b,1)(a,b,1) be a trio. Show that there exist sequence (xn),(yn)(x_n),(y_n), and (zn)(z_n) converging respectively to a,ba,b, and cc such that (xn,yn,1)(x_n,y_n,1) is a rational trio for all nn. (i) Let (a,b,1)(a,b,1) be a reduced trio. Show that there exist sequence (xn),(yn)(x_n),(y_n), and (zn)(z_n) converging respectively to a,ba,b, and cc such that (xn,yn,zn)(x_n,y_n,z_n) is a rational reduced trio for all nn.
[hide=Part 3] Denote j=e2iπ/3=−12+i32j=e^{2i\pi/3}=-\frac12+i\frac{\sqrt3}2. For each trio T=(a,b,c)T=(a,b,c) we define T=(a,c,b)\mathcal T=(a,c,b), S(T)=a+b+cS(T)=a+b+c and z(T)=a+bj+cj2z(T)=a+bj+cj^2.
(a) Express the modulus of z(T)z(T) as a function of S(T)S(T). Can we have z(T)=0z(T)=0? Compute the sine and the cosine of the argument θ\theta of z(T)z(T) in terms of a,b,ca,b,c. (b) Let z0z_0 be a given nonzero complex number. Find all trios T=(a,b,c)T=(a,b,c) such that z(T)=z0z(T)=z_0. (c) Given trios T1T_1 and T2T_2, prove that there is a unique trio, to be denoted as T1∗T2T_1*T_2, satisfying S(T1∗T2)=S(T1)S(T2)S(T_1*T_2)=S(T_1)S(T_2) and z(T1∗T2)=z(T1)z(T2)z(T_1*T_2)=z(T_1)z(T_2). Compute T1∗T2T_1*T_2 in terms of T1T_1 and T2T_2. What can be said about the argument of z(T1∗T2)z(T_1*T_2)? What can be said about z(T1∗T1)z(T_1*\mathcal T_1)? (d) If T1T_1 and T2T_2 are reduced trios, is T1∗T2T_1*T_2? The same question if the word ”reduced” is replaced by ”integer” and by ”primitive”. (e) Compare the trios T1∗T2T_1*T_2 and T2∗T1T_2*T_1; T1∗(T2∗T3)T_1*(T_2*T_3) and (T1∗T2)∗T3(T_1*T_2)*T_3, T1T_1 and T1∗(1,0,0)T_1*(1,0,0). (f) Given trios T1T_1 and T2T_2, solve the equation T1∗T=T2T_1*T=T_2 in TT. (g) Given a trio TT, define the sequence of trios (Tn)(T_n) by T0=(1,0,0)T_0=(1,0,0) and Tn+1=T∗TnT_{n+1}=T*T_n. Calculate S(Tn)S(T_n). Given an integer pp, find all TT for which Tp=T0T_p=T_0.
[hide=Part 4] Denote by AA the set of integers mm that are of the form u2+3v2u^2+3v^2, where u,vu,v are integers. Denote by A’A’ the set of nonzero complex numbers z=u+iv3z=u+iv\sqrt3, where u,vu,v are integers (note that ∣z∣2=u2+3v2|z|^2=u^2+3v^2). Denote by BB the set of nonzero integers nn of the form r2+rs+s2r^2+rs+s^2, where r,sr,s are integers.
(a) Prove that a product of two elements of A’A’ belongs to A’A’, and that a product of two elements of AA belongs to AA. (b) Show that if p∈Ap\in A is a prime number, then p=3p=3 or 3∣p−13\mid p-1. (c) Prove that A=BA=B (you may note that r2+rs+s2=(r+s)2−(r+s)s+s2r^2+rs+s^2=(r+s)^2-(r+s)s+s^2). (d) Prove that every even element of AA is divisible by 44 and that its quarter belongs to AA; then prove that each element of AA is the product of a power of 44 and an odd element of AA. (e) i. Suppose that there is an odd integer m=u2+3v2m=u^2+3v^2, where u,vu,v are coprime integers, and there exists a prime divisor pp of mm not belonging to AA. Prove that there exists the smallest positive integer n0n_0 such that n0pn_0p is in AA, and that n0n_0 is odd. ii. Verify the existence of integers u’,v’u’,v’ less than p2\frac p2 in absolute value such that u−u’u-u’ and v−v’v-v’ are divisible by pp. Prove that pp divides the nonzero number u’2+3v’2u’^2+3v’^2 and hence that n0<pn_0<p. iii. Verify the existence of coprime nonzero integers u0,v0u_0,v_0 such that n0p=u02+3v02n_0p=u_0^2+3v_0^2. iv. Verify the existence of integers u1,v1u_1,v_1 less than n02\frac{n_0}2 in absolute value such that u1−u0u_1-u_0 and v1−v0v_1-v_0 are divisible by nn. Prove that n0n_0 divides the nonzero integer u12+3v12u_1^2+3v_1^2 which we’ll denote by n0n1n_0n_1. v. Deduce that such an integer mm cannot exist (you may consider the number n02n1pn_0^2n_1p). (f) Prove that every element of AA can be written in the form m=C2p1⋯pkm=C^2p_1\cdots p_k, where CC is a positive integer and pip_i distinct prime factors of AA. (g) i. Let pp be a prime number with 3∣p−13\mid p-1, and KK the set of triples (x,y,z)(x,y,z) of integers with 0<x,y,z<p0<x,y,z<p such that p∣xyz−1p\mid xyz-1. Prove that kk has exactly (p−1)2(p-1)^2 elements and that the number of those with x,y,zx,y,z not equal is divisible by 33. (h) Let DD be the set of integers dd for which there is an integer trio (a,b,c)(a,b,c) satisfying a+b+c=da + b + c = d and abc≠0abc\ne 0. Prove, using question (e) of part 22, that every element of DD has a prime divisor in AA. Conversely, what can be said about the nonzero integers having a prime divisor in AA? (i) Find the elements of DD between 20012001 and 20102010 inclusive.
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