add4-integer-method-linear-y

Additions Chapter IV — General Integer Method when One Variable is Linear

Summary: Lagrange’s fourth Addition treats general polynomial Diophantine equations of the form $a+bx+cy+d{x}^2+exy+\cdots =0$ in which $y$ appears at most to the first degree. Solving for $y$ and using a polynomial elimination of $x$ between numerator and denominator, one shows that the denominator must divide a fixed integer $A$ — so only finitely many candidate $x$-values exist, except in the special case of a constant denominator (where the residue-class machinery of Add. III applies).

Sources: additions-4

Last updated: 2026-05-10

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Position in the Additions (Appendix to Chapter III)

This chapter occupies Articles 46–48 (pp. 467–469). It is the Appendix to Chapter III of the Additions, extending the linear method to a polynomial setting.

Whereas Add. III handled $ax-by=c$ — first-degree, one unknown linear in each variable — Add. IV addresses any polynomial equation in two variables provided one variable, say $y$, has degree at most 1.

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The General Setup (Art. 46)

Equation:

$a+bx+cy+d{x}^2+exy+g{x}^{2}y+f{x}^3+h{x}^4+k{x}^{3}y+\cdots =0$

with integer coefficients $a,b,c,d,\dots$, and we seek integer solutions $(x,y)$.

Solving for $y$:

$y=-\frac{a+bx+d{x}^2+f{x}^3+h{x}^4+\cdots }{c+ex+g{x}^2+k{x}^3+\cdots }=-\frac{p(x)}{q(x)}.$

The question reduces to: for which integer $x$ does $q(x)\mid p(x)$?

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The Resultant Criterion (Art. 46, continued)

Step 1: Eliminate $x$ between $p(x)$ and $q(x)$ by the standard algebraic resultant — this gives a polynomial relation in $p,q$ alone:

$A+Bp+Cq+D{p}^2+Epq+F{q}^2+G{p}^3+\cdots =0$

with rational-integer coefficients $A,B,C,\dots$ depending on $a,b,c,d,\dots$. The constant term $A={\mathrm{Res}}_x(p,q)$ is the resultant of $p$ and $q$.

Step 2: Substitute $p=-qy$ (since $y=-p/q$):

$A-Byq+Cq+D{y}^2{q}^2-Ey{q}^2+F{q}^2+\cdots =0.$

Step 3: Every term except the leading $A$ contains a factor of $q$. Hence

$\boxed{q\mid A}.$

Algorithm:

1. Compute the resultant $A={\mathrm{Res}}_x(p,q)$ — a fixed integer.
2. Enumerate the divisors of $A$ (positive and negative). Each divisor is a candidate value of $q=q(x)$.
3. For each candidate $q$, solve the polynomial equation $q(x)=q_{\text{cand}}$ for integer $x$.
4. Substitute each integer $x$ into $p(x)/q(x)$ to check whether $y$ comes out as an integer.

Conclusion: This procedure finds all integer solutions, and the number of integer solutions is finite (since $A$ has finitely many divisors and each candidate $q$ has finitely many integer roots).

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Exception: Constant Denominator (Art. 47)

The argument fails if $q(x)$ is a constant — i.e. if all the $y$-coefficients $e,g,k,\dots$ vanish. Then

$y=-\frac{a+bx+d{x}^2+f{x}^3+h{x}^4+\cdots }{c}$

and the question becomes: for which integers $x$ is $a+bx+d{x}^2+\cdots$ divisible by $c$?

Periodicity: If $n$ satisfies the divisibility condition, so does $n\pm \mu c$ for any $\mu \in \mathbb{Z}$ (since the polynomial is integer-valued and shifting by a multiple of $c$ preserves residues mod $c$). Hence one need only check residues $|n|\le c/2$.

Solution form: If $n^{\prime },n^{\prime \prime },n^{\prime \prime \prime },\dots$ are the residues (with $|n^{(i)}|\le c/2$) that make the polynomial divisible by $c$, then all integer solutions are

$x\in \{n^{\prime }\pm {\mu }^{\prime }c,n^{\prime \prime }\pm {\mu }^{\prime \prime }c,n^{\prime \prime \prime }\pm {\mu }^{\prime \prime \prime }c,\dots :{\mu }^{(i)}\in \mathbb{Z}\}.$

Each $n^{(i)}$ generates an arithmetic progression of solutions — so the solution set is infinite in this exceptional case.

Lagrange notes that further refinements appear in his 1768 Berlin memoir Nouvelle Méthode pour résoudre les Problèmes Indéterminés — see the next chapter (Add. V) for the principal application.

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Lemma — Integrality of a Homogeneous Form (Art. 48)

Statement: Determine integers $x,y$ such that

$\frac{a{y}^m+b{y}^{m-1}x+d{y}^{m-2}{x}^2+f{y}^{m-3}{x}^3+\cdots }{c}$

is an integer.

Hypotheses: $\gcd (x,y)=1$ and $\gcd (y,c)=1$ (the latter is restrictive; relaxation discussed below).

Reduction trick: By Add. III, we can always solve $x=ny-cz$ for integers $n,z$ given $\gcd (y,c)=1$. Substituting:

$a{y}^m+b{y}^{m-1}(ny-cz)+d{y}^{m-2}(ny-cz{)}^2+\cdots$ $=(a+bn+d{n}^2+f{n}^3+\cdots )y^m-(b+2dn+3f{n}^2+\cdots )c{y}^{m-1}z+(d+3fn+\cdots )c^2{y}^{m-2}{z}^2-\cdots$

All terms except the leading one carry a factor of $c$. Since $\gcd (y,c)=1$, the form is divisible by $c$ iff the leading coefficient is:

$c\mid a+bn+d{n}^2+f{n}^3+\cdots$

This is a single-variable polynomial congruence $\mathrm{mod}c$, exactly the situation of Art. 47. Solve it for the residues $n^{\prime },n^{\prime \prime },\dots$ and recover

$x=ny-cz,z\in \mathbb{Z}.$

Generalization: If $\gcd (x,y)=\alpha \ne 1$, set $x=\alpha x^{\prime }$, $y=\alpha y^{\prime }$ with $\gcd (x^{\prime },y^{\prime })=1$ and proceed; if $\gcd (y,c)=\beta \ne 1$, set $y=\beta y^{\prime \prime }$ with $\gcd (y^{\prime \prime },c)=1$. The lemma then applies in the reduced variables.

This lemma is the key technical input to Add. V — see add5-rational-quadratic-surds Art. 51–52, which uses it to reduce the problem of $\sqrt{A{y}^2+B}$ rational to a residue search modulo $A$.

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Comparison with the Main Text

This chapter has no direct counterpart in Euler’s main Elements of Algebra, which treats specific Diophantine families (linear, Pell, surd-rationalization) but not the general polynomial case where one variable is linear.

The closest main-text relatives are:

• ch2.0.3-compound-indeterminate-equations — compound equations where $y$ appears linearly and $x$ to higher degrees; Lagrange’s Art. 46 generalizes Euler’s heuristic divisor scan.
• ch2.0.4-surd-rationalization — rationalization of $\sqrt{a+bx+c{x}^2}$, the special case treated in Add. V.

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Related pages

• add3-integer-linear-equations — the linear case used as a building block (Art. 48 substitution)
• add5-rational-quadratic-surds — the principal application: Lagrange’s quadratic-surd algorithm
• ch2.0.3-compound-indeterminate-equations — Euler’s main-text version of the same problem class
• indeterminate-analysis — overview of integer-Diophantine techniques
• linear-diophantine-equations — concept page used by Art. 48