![optic_equation_integer_solutions](https://upload.wikimedia.org/wikipedia/commons/5/59/Optic_equation_integer_solutions.svg)
In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, the optic equation is an equation that requires the sum of the
reciprocals of two positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':
[Dickson, L. E., ''History of the Theory of Numbers, Volume II: Diophantine Analysis'', Chelsea Publ. Co., 1952, pp. 688–691.]
:
Multiplying both sides by ''abc'' shows that the optic equation is equivalent to a
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
(a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
in multiple integer variables).
Solution
All solutions in integers ''a, b, c'' are given in terms of positive integer parameters ''m, n, k'' by
[
:
where ''m'' and ''n'' are ]coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
.
Appearances in geometry
The optic equation, permitting but not requiring integer solutions, appears in several contexts in geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
.
In a bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' r ...
, the inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
''r'', the circumradius ''R'', and the distance ''x'' between the incenter and the circumcenter are related by Fuss' theorem according to
:
and the distances of the incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...
''I'' from the vertices ''A, B, C, D'' are related to the inradius according to
:
In the crossed ladders problem The crossed ladders problem is a puzzle of unknown origin that has appeared in various publications and regularly reappears in Web pages and Usenet discussions.
The problem
Two ladders of lengths ''a'' and ''b'' lie oppositely across an alley, a ...
, two ladders braced at the bottoms of vertical walls cross at the height ''h'' and lean against the opposite walls at heights of ''A'' and ''B''. We have Moreover, the formula continues to hold if the walls are slanted and all three measurements are made parallel to the walls.
Let P be a point on the circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of an equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
''ABC'', on the minor arc ''AB''. Let ''a'' be the distance from ''P'' to ''A'' and ''b'' be the distance from ''P'' to ''B''. On a line passing through ''P'' and the far vertex ''C'', let ''c'' be the distance from ''P'' to the triangle side ''AB''. Then[Posamentier, Alfred S., and Salkind, Charles T., ''Challenging Problems in Geometry'', Dover Publ., 1996.]
In a trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a Convex polygon, convex quadri ...
, draw a segment parallel to the two parallel sides, passing through the intersection of the diagonals and having endpoints on the non-parallel sides. Then if we denote the lengths of the parallel sides as ''a'' and ''b'' and half the length of the segment through the diagonal intersection as ''c'', the sum of the reciprocals of ''a'' and ''b'' equals the reciprocal of ''c''.
The special case in which the integers whose reciprocals are taken must be square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s appears in two ways in the context of right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s. First, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse. This holds whether or not the numbers are integers; there is a formula (see here
Here is an adverb that means "in, on, or at this place". It may also refer to:
Software
* Here Technologies, a mapping company
* Here WeGo (formerly Here Maps), a mobile app and map website by Here
Television
* Here TV (formerly "here!"), a TV ...
) that generates all integer cases. Second, also in a right triangle the sum of the squared reciprocal of the side of one of the two inscribed squares and the squared reciprocal of the hypotenuse equals the squared reciprocal of the side of the other inscribed square.
The sides of a heptagonal triangle
A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter an ...
, which shares its vertices with a regular heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon.
The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than ''hepta-'', a Greek-derived num ...
, satisfy the optic equation.
Other appearances
Thin lens equation
For a lens of negligible thickness and focal length ''f'', the distances from the lens to an object, ''S''1, and from the lens to its image, ''S''2, are related by the thin lens formula
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
:
: .
Electrical engineering
Components of an electrical circuit or electronic circuit can be connected in what is called a series or parallel configuration. For example, the total resistance value ''Rt'' of two resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s with resistances ''R''1 and ''R''2 connected in ''parallel'' follows the optic equation:
: .
Similarly, the total inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
''Lt'' of two inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s with inductances ''L''1 and ''L''2 connected in ''parallel'' is given by:
:
and the total capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
''Ct'' of two capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s with capacitances ''C''1 and ''C''2 connected in ''series'' is as follows:
: .
Paper folding
The optic equation of the crossed ladders problem can be applied to folding rectangular paper into three equal parts. One side (the left one illustrated here) is partially folded in half and pinched to leave a mark. The intersection of a line from this mark to an opposite corner, with a diagonal is exactly one third from the bottom edge. The top edge can then be folded down to meet the intersection.[; see in particular section]
Dividing into thirds
Harmonic mean
The harmonic mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
of ''a'' and ''b'' is or 2''c''. In other words, ''c'' is half the harmonic mean of ''a'' and ''b''.
Relation to Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
states that the sum of two integers each raised to the same integer power ''n'' cannot equal another integer raised to the power ''n'' if ''n'' > 2. This implies that no solutions to the optic equation have all three integers equal to perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...
s with the same power ''n'' > 2. For if then multiplying through by would give which is impossible by Fermat's Last Theorem.
See also
*Erdős–Straus conjecture
The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer n that is 2 or more, there exist positive integers x, y, and z for which \frac=\frac+\frac+\frac.
In other words, the number 4/n ...
, a different Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
involving sums of reciprocals of integers
*Sums of reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many n ...
References
{{reflist
Diophantine equations