Indeterminate Analysis

Summary: The branch of algebra that studies equations with fewer equations than unknowns, seeking solutions restricted to integer (or rational) values; the restriction to integers typically limits the solution set to finitely many or to an arithmetic progression.

Sources: chapter-2.0.1, chapter-2.0.2, chapter-2.0.3, chapter-2.0.4, chapter-2.0.5, chapter-2.0.6, chapter-2.0.7, chapter-2.0.8, chapter-2.0.9, chapter-2.0.10, chapter-2.0.11, chapter-2.0.12, chapter-2.0.13, chapter-2.0.14, chapter-2.0.15, additions-3, additions-4, additions-5, additions-6

Last updated: 2026-05-10

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Definition

A question is indeterminate when it does not furnish as many equations as there are unknown quantities to be determined; some unknowns then remain free. In pure algebra this freedom yields infinitely many solutions over the reals, but annexing the condition that all values must be integer and positive (or at least rational) drastically reduces the solution count. (source: chapter-2.0.1, §1–2)

Scope and Difficulty

• Sometimes only finitely many integer solutions exist.
• Sometimes infinitely many exist but are not easily enumerated.
• Sometimes no integer solution exists at all.

Because of this variability, Euler notes that the subject “frequently requires artifices entirely appropriate to it, which are of great service in exercising the judgment of beginners.” (source: chapter-2.0.1, §2)

Main Problem Classes (Euler’s Part II)

Chapter	Form	Method
I	$ax+by=c$, one equation, two unknowns	Iterative remainder reduction; Euclidean-table back-substitution
II (Regula Caeci)	Two equations, three or more unknowns	Eliminate one variable; reduce to Chapter I
III	$xy+\text{(linear terms)}=c$; $y$ linear, $x$ higher	Divisor condition on a known constant
IV	$a+bx+c{x}^2$ must be a perfect square	Four rationalization rules; bootstrap from a known solution
V	$a{t}^2+b{u}^2$ impossibility	Residue-class analysis mod 3, 4, 5, 7; quadratic residues
VI	$a{x}^2+b=y^2$ in integers	Seed solution + Pell pair $m^2=a{n}^2+1$ yields infinite integer families
VII	Solving $m^2=a{n}^2+1$ in integers	Pell’s descent method; closed forms for $a$ near a square; table for $a\le 100$
VIII	$a+bx+c{x}^2+d{x}^3$ must be a perfect square	Two-term and three-term root ansätze when $a=f^2$; bootstrap; one new $x$ per pass
IX	$a+bx+c{x}^2+d{x}^3+e{x}^4$ must be a perfect square	Three subclasses (first end square, last end square, both); up to six new values; degree 4 is the upper limit (§146)
X	$a+bx+c{x}^2+d{x}^3$ must be a perfect cube	Three subclasses by which end is a cube; biquadrate sketch; sum-of-two-cubes impossibility
XI	Factorization of $a{x}^2+bxy+c{y}^2$	Imaginary factorization; Brahmagupta-Fibonacci identity $(p^2+c{q}^2)(r^2+c{s}^2)=x^2+c{y}^2$; proto-genus theory I × I = II, etc.
XII	$a{x}^2+c{y}^2$ as a square, cube, or higher power	Set $x\sqrt{a}+y\sqrt{-c}=(p\sqrt{a}+q\sqrt{-c}{)}^n$; odd powers always solvable; even powers need a seed
XIII	Impossibility of $x^4\pm y^4=\square$, $x^4+2y^4=\square$, etc.	Infinite descent (Fermat’s method); FLT for $n=4$; contrast with $x^4-2y^4$ which has infinitely many solutions
XIV	Seventeen showcase problems: $x\pm a$, $x^2+y$, $xy+1$ etc. all squares	Application of all prior artifices; theorem $a\pm x$ both squares ⇔ $a$ is sum of two squares; impossibility of $p^2+q^2\&p^2+3q^2$ both squares (descent)
XV	Cube-formula problems; $x^3+y^3+z^3=v^3$; AP-cube questions	FLT for $n=3$ via $(t+u\sqrt{-3}{)}^3$ + descent; theorem $x^3\pm y^3\ne 2z^3$; four-parameter family for three-cubes-as-cube

Lagrange’s Additions III–VI (Appendices)

Lagrange’s Additions extend the apparatus with general algorithms that subsume several of Euler’s case-by-case methods:

Chapter	Form	Method
Add. III	$ax-by=c$ in integers	CF of $b/a$; second-to-last convergent gives the seed solution; subsumes Euler’s Chapter I
Add. IV	Polynomial in $(x,y)$ with $y$ linear	Resultant divisor scan: denominator $q(x)$ must divide ${\mathrm{Res}}_x(p,q)$; finite candidate set
Add. V	$\sqrt{a+bx+c{x}^2}$ rational	Lagrange’s descent: strictly decreasing $A,A^{\prime },A^{\prime \prime },\dots$ via $z=nq-A{q}^{\prime }$; first uniform decision procedure; subsumes Chapters IV
Add. VI	Double and triple equalities	Linear cases reduce to a simple equality (Add. V); quadratic doubles produce a quartic with no general method

Impossibility

For $bp=aq+n$ to be solvable in integers, $\gcd (a,b)$ must divide $n$. If not, the iterative reduction produces a permanent non-integer, proving impossibility. (source: chapter-2.0.1, §22–23)

Solution Structure

When solutions exist and are infinite in number, they form an arithmetical progression: if $N_0$ is one solution to $ax-by=c$, all solutions are $N_0+k\cdot ab/\gcd (a,b)$ for $k\in \mathbb{Z}$. The common difference equals the product of the (reduced) moduli. (source: chapter-2.0.1, §13)

Related pages

• linear-diophantine-equations
• regula-caeci
• greatest-common-divisor
• arithmetical-progressions
• rationalization
• pythagorean-triples
• ch2.0.1-indeterminate-equations-first-degree
• ch2.0.2-regula-caeci
• ch2.0.3-compound-indeterminate-equations
• ch2.0.4-surd-rationalization
• ch2.0.5-impossibility-quadratic-squares
• ch2.0.6-integer-solutions-quadratic-squares
• ch2.0.7-pell-equation-method
• ch2.0.8-cubic-surd-rationalization
• ch2.0.9-quartic-surd-rationalization
• ch2.0.10-cubic-formula-as-cube
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• ch2.0.13-impossibility-biquadrate-sums
• ch2.0.14-questions-squares
• ch2.0.15-questions-cubes
• quadratic-residues
• pell-equation
• sum-of-two-cubes
• brahmagupta-fibonacci-identity
• sums-of-two-squares
• fermats-last-theorem-n3
• fermats-last-theorem-n4
• three-cubes-as-cube
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• add5-rational-quadratic-surds
• add6-double-triple-equalities
• lagrange-reduction-algorithm
• double-and-triple-equalities