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Mathematics
V. P. MYAKISHEV
ON A DIOPHANTINE EQUATION OF THE THIRD DEGREE
(Presented by Academician I. M. Vinogradov on 4 IX 1962)
The present article is a continuation of the work \((^1)\). Here we give a formula for finding all primitive solutions, i.e., solutions in relatively prime numbers, of a Diophantine equation of the third degree, and also an asymptotic formula for the number of primitive integral points lying on a certain two-dimensional surface of the third order in four-dimensional space.
Consider the Diophantine equation
\[ \left| \begin{array}{ccc} x & y & z\\ -7z & x+7z & y\\ -7y & 7y-7z & x+7z \end{array} \right|=t^3 \tag{1} \]
and let us find all its primitive solutions, i.e., solutions for which
\(\gcd(x,y,z,t)=1\). Define the following four forms in three variables \(m,n\), and \(l\):
\[ \begin{aligned} X_1(m,n,l)={}&98l^3+m^3-42mn^2+98ml^2-14m^2n+28m^2l-28n^3\\ &{}-49n^2l+49nl^2,\\ X_2(m,n,l)={}&-49l^3-35ml^2-7m^2l+6m^2n+14mn^2+14mnl\\ &{}+7n^3+28n^2l,\\ X_3(m,n,l)={}&7n^3-7nl^2+7n^2l+9mn^2-3m^2l-7ml^2+3m^2n,\\ X_4(m,n,l)={}&49l^3+m^3-7mn^2+49ml^2+14m^2l+21mnl-7n^3+49nl^2. \end{aligned} \]
For integral \(m,n,l\), put
\[ D=\gcd\bigl(X_1(m,n,l),\ X_2(m,n,l),\ X_3(m,n,l)\bigr). \]
Theorem 1. All primitive solutions of the Diophantine equation \((1)\) are obtained exactly once from the formulas
\[ x=\frac{X_1(m,n,l)}{D},\qquad y=\frac{X_2(m,n,l)}{D},\qquad z=\frac{X_3(m,n,l)}{D},\qquad t=\frac{X_4(m,n,l)}{D}, \]
where \(m,n,l\) run through all triples of integers subject to the condition
\(\gcd(m,n,l)=1\).
Proof. We shall only outline the proof.
Lemma 1. All solutions in rational numbers of the equation
\[ \left| \begin{array}{ccc} x_1 & x_2 & x_3\\ -7x_3 & x_1+7x_3 & x_2\\ -7x_2 & 7x_2-7x_3 & x_1+7x_3 \end{array} \right|=1 \tag{2} \]
are obtained exactly once when, in the formulas
\[ x_1=\frac{X_1(m,n,l)}{X_4(m,n,l)},\qquad x_2=\frac{X_2(m,n,l)}{X_4(m,n,l)},\qquad x_3=\frac{X_3(m,n,l)}{X_4(m,n,l)}, \]
the parameters \(m,n,l\) run through all integers satisfying the conditions:
1) either \(m>0,\ \gcd(m,n,l)=1\);
2) or \(m=0,\ n>0,\ \gcd(m,n,l)=1\);
3) \(m=0,\ n=0,\ l=1\).
Lemma 2.
\[ \gcd\bigl(X_1(m,n,l),\ X_2(m,n,l),\ X_3(m,n,l)\bigr) \]
\[ =\gcd\bigl(X_1(m,n,l),\ X_2(m,n,l),\ X_3(m,n,l),\ X_4(m,n,l)\bigr). \]
Without proving these lemmas, let us only note that in proving them we used the close connection of equation (2) with the cyclic cubic extension \(R(\alpha)\) of the field of rational numbers \(R\) (\(\alpha\) is a root of the equation irreducible over \(R\),
\(x^3-7x+7=0\)).
It is clear that our theorem follows from the indicated lemmas.
Remark. The quantity \(D\) occurring in the formulation of the theorem can be given another interpretation. Denote by \(\Omega\) the ring of integers of the field \(R(\alpha)\). Let \(B=m+n\alpha+l\alpha^2\) be an integer of the field \(R(\alpha)\), with \(\gcd(m,n,l)=1\). The decomposition of the ideal \((B)\) (if \(B\) is not a unit of the ring \(\Omega\)) into prime ideals can be written in the form
\[ (B)=(\pi_7)^\nu \prod_s(\pi_s)^{k_s}(\sigma\pi_s)^{k'_s}, \]
where \(\nu=0,1\), or \(2\), \(k_s>0\), \(k'_s\ge 0\), and \(\sigma\) is a generating automorphism of the Galois group of the field \(R(\alpha)\). Then
\[ D= \begin{cases} 1, & \text{if } B \text{ is a unit of the ring } \Omega,\\[6pt] 7^\nu \displaystyle\prod_{\substack{s\\ k'_s>0}} (\operatorname{Norm}\pi_s)^{\min(k_s,2k'_s)}, & \text{if } B \text{ is not a unit of the ring } \Omega. \end{cases} \]
From this remark it is seen that \(D\) may grow without bound as \(m,n,l\) vary. This circumstance makes the study of the distribution of integral points on the surface (1) difficult. However, for one case we can solve the corresponding distribution problem.
Denote by \(F(h)\) the number of primitive integral points of the form \((l_1,l_2,l_3,l_4)\) with \(l_4>0\), lying on the surface given parametrically by
\[ x_1=14y_1^3+7y_1^2y_2-7y_1y_2^2-4y_2^3, \]
\[ x_2=-7y_1^3+4y_1^2y_2+y_2^3, \]
\[ x_3=-y_1^2y_2+y_1y_2^2+y_2^3, \]
\[ x_4=7y_1^3+7y_1^2y_2-y_2^3, \]
where \(y_1\) and \(y_2\) run through real numbers, \(y_2>0\), and \(y_1^2+y_2^2\le h\).
Theorem 2. As \(h\to\infty\),
\[ F(h)=\frac{\theta}{16\pi^2}\,h+O(\sqrt h\ln h), \]
where
\[ \theta= \frac{48\sqrt[3]{49}-1}{7} \arctg \frac{7\,[k_3(k_2-k_1)+49+k_1k_2]} {(49+k_1k_2)k_3-49(k_2-k_1)} + 49\arctg \frac{k_3(k_2-k_1)+k_1k_2+1} {k_3(1+k_1k_2)+k_1-k_2}. \]
Here \(k_1,k_2,k_3\) are the roots of the equation
\[ 7z^3+7z^2-1=0,\qquad k_1<0,\quad k_2<0,\quad k_3>0,\quad |k_1|>|k_2|. \]
V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
30 VIII 1962
REFERENCES
- V. P. Myakishev, DAN, 143, No. 4, 785 (1962).