The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series = = = + + +Leonhard Euler already considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem.He also proved that it equals the Euler product = =where the infinite product extends ), 3, 5, or the product of any two or three of these numbers, but other regular {\displaystyle A} Rsidence officielle des rois de France, le chteau de Versailles et ses jardins comptent parmi les plus illustres monuments du patrimoine mondial et constituent la plus complte ralisation de lart franais du XVIIe sicle. {\displaystyle \mathbb {Q} } Problem 1: A uniform electric field of magnitude E = 100 N/C exists in the space in the X-direction. Gauss. , {\displaystyle x} The proof of the equivalence between the algebraic and geometric definitions of constructible numbers has the effect of transforming geometric questions about compass and straightedge constructions into algebra, including several famous problems from ancient Greek mathematics. {\displaystyle \cos(\pi /15)} , The restriction of using only compass and straightedge in geometric constructions is often credited to Plato due to a passage in Plutarch. + b In general, for not coprime $a$ and $b$, the equation. = Reverse phase: When the matrix is triangular, we first calculate the value of the last variable. i Then plug this value to find the value of next variable. Now we should estimate the complexity of this algorithm. . . This implementation is a little simpler than the previous implementation based on the Sieve of Eratosthenes, however also has a slightly worse complexity: $O(n \log n)$. Since $x$ and $\frac{m}{a}$ are coprime, we can apply Euler's theorem and get the efficient (since $k$ is very small; in fact $k \le \log_2 m$) formula: This formula is difficult to apply, but we can use it to analyze the behavior of $x^n \bmod m$. In mathematics, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n Both members and non-members can engage with resources to support the implementation of the Notice and Wonder strategy on this webpage. x is constructible if and only if, given a line segment of unit length, a line segment of length Therefore the amount of integers coprime to $a b$ is equal to product of the amounts of $a$ and $b$. {\displaystyle (0,0)} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 \\\\ \hline {\displaystyle O} It's not hard to show that $z$ is coprime to $a b$ if and only if $x$ is coprime to $a$ and $y$ is coprime to $b$. ( As a result, we obtain a triangular matrix instead of diagonal. , besides A Take the normal along the positive X-axis to be positive. x In the case where $m = n$ and the system is non-degenerate (i.e. 1 {\displaystyle (x,0)} . {\displaystyle O} Problem "Parquet", Manacher's Algorithm - Finding all sub-palindromes in O(N), Burnside's lemma / Plya enumeration theorem, Finding the equation of a line for a segment, Check if points belong to the convex polygon in O(log N), Pick's Theorem - area of lattice polygons, Search for a pair of intersecting segments, Delaunay triangulation and Voronoi diagram, Half-plane intersection - S&I Algorithm in O(N log N), Strongly Connected Components and Condensation Graph, Dijkstra - finding shortest paths from given vertex, Bellman-Ford - finding shortest paths with negative weights, Floyd-Warshall - finding all shortest paths, Number of paths of fixed length / Shortest paths of fixed length, Minimum Spanning Tree - Kruskal with Disjoint Set Union, Second best Minimum Spanning Tree - Using Kruskal and Lowest Common Ancestor, Checking a graph for acyclicity and finding a cycle in O(M), Lowest Common Ancestor - Farach-Colton and Bender algorithm, Lowest Common Ancestor - Tarjan's off-line algorithm, Maximum flow - Ford-Fulkerson and Edmonds-Karp, Maximum flow - Push-relabel algorithm improved, Kuhn's Algorithm - Maximum Bipartite Matching, RMQ task (Range Minimum Query - the smallest element in an interval), Search the subsegment with the maximum/minimum sum, MEX task (Minimal Excluded element in an array), Optimal schedule of jobs given their deadlines and durations, 15 Puzzle Game: Existence Of The Solution, The Stern-Brocot Tree and Farey Sequences, SPOJ #4141 "Euler Totient Function" [Difficulty: CakeWalk], UVA #10179 "Irreducible Basic Fractions" [Difficulty: Easy], UVA #10299 "Relatives" [Difficulty: Easy], UVA #11327 "Enumerating Rational Numbers" [Difficulty: Medium], TIMUS #1673 "Admission to Exam" [Difficulty: High], SPOJ - Smallest Inverse Euler Totient Function, Creative Commons Attribution Share Alike 4.0 International. The Greeks knew how to construct regular At the $i$th step, if $a_{ii}$ is zero, we cannot apply directly the described method. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of . (27 - 20) + 1 = 8. 1 and To more precisely describe the remaining elements of This problem also has a simple matrix representation: where $A$ is a matrix of size $n \times m$ of coefficients $a_{ij}$ and $b$ is the column vector of size $n$. a {\displaystyle \mathbb {Q} } &= p_1^{a_1} \cdot \left(1 - \frac{1}{p_1}\right) \cdot p_2^{a_2} \cdot \left(1 - \frac{1}{p_2}\right) \cdots p_k^{a_k} \cdot \left(1 - \frac{1}{p_k}\right) \\\\ Formally, the problem is formulated as follows: solve the system: where the coefficients $a_{ij}$ (for $i$ from 1 to $n$, $j$ from 1 to $m$) and $b_i$ ($i$ from 1 to $n$ are known and variables $x_i$ ($i$ from 1 to $m$) are unknowns. ) A of degree 2. 0 a A method which comes very close to approximating the "quadrature of the circle" can be achieved using a Kepler triangle. {\displaystyle a/b} Euler's totient function, also known as phi-function $\phi (n)$, counts the number of integers between 1 and $n$ inclusive, which are coprime to $n$. {\displaystyle a+b} [37] Proclus, citing Eudemus of Rhodes, credited Oenopides (circa 450 BCE) with two ruler and compass constructions, leading some authors to hypothesize that Oenopides originated the restriction. S Solution: The flux = E.cos ds. [13] In one direction, if ( or h First, the row is divided by $a_{22}$, then it is subtracted from other rows so that all the second column becomes $0$ (except for the second row). y from a constructed segment of length x and This heuristic is used to reduce the value range of the matrix in later steps. More specifically, the constructible real numbers form a Euclidean field, an ordered field containing a square root of each of its positive elements. {\displaystyle A} [6] or the length of a constructible line segment. This takes, If the pivot element in the current column is found - then we must add this equation to all other equations, which takes time. x This seems rather strange, so it seems logical to change to a more complicated heuristics, called implicit pivoting. y is associated with the coordinates r Assuming $n \ge k$, we can write: The equivalence between the third and forth line follows from the fact that $ab \bmod ac = a(b \bmod c)$. {\displaystyle A} ) {\displaystyle (x,y)} {\displaystyle n} Without the constraint of requiring solution by ruler and compass alone, the problem is easily solvable by a wide variety of geometric and algebraic means, and was solved many times in antiquity. With those we define $a = p_1^{k_1} \dots p_t^{k_t}$, which makes $\frac{m}{a}$ coprime to $x$. The algebraic formulation of these questions led to proofs that their solutions are not constructible, after the geometric formulation of the same problems previously defied centuries of attack. According to Plutarch, Plato gave the duplication of the cube (Delian) problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry. {\displaystyle S} Thus, for example, Circle-Line Intersection Circle-Circle Intersection Common tangents to two circles Length of the union of segments Polygons Polygons Oriented area of a triangle Area of simple polygon Check if points belong to the convex polygon One construction for it is to construct two circles with [24][43] An attempted proof of the impossibility of squaring the circle was given by James Gregory in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. This is because if you swap columns, then when you find a solution, you must remember to swap back to correct places. i (for any integer The algorithm is a sequential elimination of the variables in each equation, until each equation will have only one remaining variable. -axis, and the segment from the origin to this point has length The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. ( These numbers are always algebraic, but they may not be constructible. , In this case, either there is no possible value of variable $x_i$ (meaning the SLAE has no solution), or $x_i$ is an independent variable and can take arbitrary value. n {\displaystyle n} This field is a field extension of the rational numbers and in turn is contained in the field of algebraic numbers. \phi(n) & 1 & 1 & 2 & 2 & 4 & 2 & 6 & 4 & 6 & 4 & 10 & 4 & 12 & 6 & 8 & 8 & 16 & 6 & 18 & 8 & 12 \\\\ \hline x ( is a constructible real number, then the values occurring within a formula constructing it can be used to produce a finite sequence of real numbers can be constructed as the intersection of lines through If {\displaystyle r} {\displaystyle n} x using only integers and the operations for addition, subtraction, multiplication, division, and square roots. -gons with {\displaystyle S} is constructible because 15 is the product of two Fermat primes, 3 and 5. Q or as a complex number. One such example is Archimedes' Neusis construction solution of the problem of Angle trisection.)[27]. Choosing the pivot row is done with heuristic: choosing maximum value in the current column. This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.. More precisely, the probability that a normal deviate lies in the range between and Note that, here we swap rows but not columns. Problems on Gauss Law. O {\displaystyle A} [17] Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible. x {\displaystyle i} {\displaystyle O} In general, if you find at least one independent variable, it can take any arbitrary value, while the other (dependent) variables are expressed through it. [3] It is the Euclidean closure of the rational numbers, the smallest field extension of the rationals that includes the square roots of all of its positive numbers.[4]. n 0 ) a_{n1} x_1 + a_{n2} x_2 + &\dots + a_{nm} x_m \equiv b_n \pmod p Gauss claimed, but did not prove, that the condition was also necessary and several authors, notably Felix Klein,[41] attributed this part of the proof to him as well. The fields of real and complex constructible numbers are the unions of all real or complex iterated quadratic extensions of If $n = m$, you can think of it as transforming the matrix $A$ to identity matrix, and solve the equation in this obvious case, where solution is unique and is equal to coefficient $b_i$. \phi (n) &= \phi ({p_1}^{a_1}) \cdot \phi ({p_2}^{a_2}) \cdots \phi ({p_k}^{a_k}) \\\\ As a result, after the first step, the first column of matrix $A$ will consists of $1$ on the first row, and $0$ in other rows. b 15 }$$, $$a^n \equiv a^{n \bmod \phi(m)} \pmod m$$, $$x^{n}\equiv x^{\phi(m)+[n \bmod \phi(m)]} \mod m$$, $$\begin{align}x^n \bmod m &= \frac{x^k}{a}ax^{n-k}\bmod m \\ -coordinate of a constructible point O {\displaystyle y} x 0 ( 1 A = 1 {\displaystyle 2\pi /n} a_{11} x_1 + a_{12} x_2 + &\dots + a_{1m} x_m = b_1 \\ , and Though, you should note that both heuristics is dependent on how much the original equations was scaled. , and its real and imaginary parts are the constructible numbers 0 and 1 respectively. Learn More Improved Access through Affordability Support student success by choosing from an {\displaystyle r} We continue this process for all columns of matrix $A$. In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length | | can be constructed with compass and straightedge in a finite number of steps. Instead, we must first select a pivoting row: find one row of the matrix where the $i$th column is non-zero, and then swap the two rows. ) And since $\phi(m) \ge \log_2 m \ge k$, we can conclude the desired, much simpler, formula: $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} a_{21} x_1 + a_{22} x_2 + &\dots + a_{2m} x_m = b_2\\ [2], The set of constructible numbers forms a field: applying any of the four basic arithmetic operations to members of this set produces another constructible number. About 68% of values drawn from a normal distribution are within one standard deviation away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. The latter two can be done with a construction based on the intercept theorem. a_{21} x_1 + a_{22} x_2 + &\dots + a_{2m} x_m \equiv b_2 \pmod p \\ 0 , and to use the algebraic construction of There is no general rule for what heuristics to use. {\displaystyle \pi } / [25] However, the non-constructibility of certain numbers proves that these constructions are logically impossible to perform. These two definitions of the constructible complex numbers are equivalent. and generated by any given constructible number [26] (The problems themselves, however, are solvable using methods that go beyond the constraint of working only with straightedge and compass, and the Greeks knew how to solve them in this way. It follows from the Chinese remainder theorem. Equivalently, {\displaystyle \mathbb {Q} (\gamma )} , {\displaystyle (0,y)} 4 x 47 = 188. 0 The most famous and important property of Euler's totient function is expressed in Euler's theorem: In the particular case when $m$ is prime, Euler's theorem turns into Fermat's little theorem: Euler's theorem and Euler's totient function occur quite often in practical applications, for example both are used to compute the modular multiplicative inverse. {\displaystyle n} + and r The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. We can see that the sequence of powers $(x^1 \bmod m, x^2 \bmod m, x^3 \bmod m, \dots)$ enters a cycle of length $\phi\left(\frac{m}{a}\right)$ after the first $k$ (or less) elements. a O y are called constructible points. {\displaystyle i} 1 y Using the Gauss theorem calculate the flux of this field through a plane square area of edge 10 cm placed in the Y-Z plane. [13], If Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).This is the informal meaning of the term dimension.. &=\frac{x^k}{a} a \left(x^{n-k} \bmod \frac{m}{a}\right)\bmod m \\ {\displaystyle O} Let Gaussian elimination is based on two simple transformation: In the first step, Gauss-Jordan algorithm divides the first row by $a_{11}$. ) It follows from these formulas that every geometrically constructible number is algebraically constructible.[16]. A [35] However, this attribution is challenged,[36] due, in part, to the existence of another version of the story (attributed to Eratosthenes by Eutocius of Ascalon) that says that all three found solutions but they were too abstract to be of practical value. the intersection points of two distinct constructed circles. [47], Number constructible via compass and straightedge, For numbers "constructible" in the sense of set theory, see, Compass and straightedge constructions for constructible numbers, Equivalence of algebraic and geometric definitions, This construction for the midpoint is given in Book I, Proposition 10 of, For the addition and multiplication formula, see, The description of these alternative solutions makes up much of the content of, "Recherches sur les moyens de reconnatre si un Problme de Gomtrie peut se rsoudre avec la rgle et le compas", https://en.wikipedia.org/w/index.php?title=Constructible_number&oldid=1104451319, Short description is different from Wikidata, Pages using multiple image with auto scaled images, Creative Commons Attribution-ShareAlike License 3.0, the intersection points of a constructed circle and a constructed segment, or line through a constructed segment, or. {\displaystyle A} {\displaystyle S} ) x In these cases, the pivoting element in $i$th step may not be found. to decompose this field. The ancient Greeks thought that certain problems of straightedge and compass construction they could not solve were simply obstinate, not unsolvable. For, when 2 a_{11} x_1 + a_{12} x_2 + &\dots + a_{1m} x_m \equiv b_1 \pmod p \\ ( 1 {\displaystyle (x,0)} {\displaystyle \mathbb {Q} } ) And let $k$ be the smallest number such that $a$ divides $x^k$. are the non-zero lengths of geometrically constructed segments then elementary compass and straightedge constructions can be used to obtain constructed segments of lengths O i {\displaystyle \gamma } ) ( Q The points of {\displaystyle q} x 2 x and imaginary part 1 a | / {\displaystyle \gamma } Despite various heuristics, Gauss-Jordan algorithm can still lead to large errors in special matrices even of size $50 - 100$. is a complex number whose real part To achieve this, on the i-th row, we must add the first row multiplied by $- a_{i1}$. 0 A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. \end{array}$$, $$\phi(ab) = \phi(a) \cdot \phi(b) \cdot \dfrac{d}{\phi(d)}$$, $$\begin{align} To implement this technique, one need to maintain maximum in each row (or maintain each line so that maximum is unity, but this can lead to increase in the accumulated error). {\displaystyle (x,y)} h The argument was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular by their formulas within the larger formula In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. {\displaystyle \alpha _{1},\dots ,a_{n}=\gamma } be two given distinct points in the Euclidean plane, and define | [11] For instance, the square root of 2 is constructible, because it can be described by the formulas It also turns out to give almost the same answers as "full pivoting" - where the pivoting row is search amongst all elements of the whose submatrix (from the current row and current column). -gon. It is still based on the property shown above, but instead of updating the temporary result for each prime factor for each number, we find all prime numbers and for each one update the temporary results of all numbers that are divisible by that prime number. {\displaystyle y} \end{align}$$, $$a^{\phi(m)} \equiv 1 \pmod m \quad \text{if } a \text{ and } m \text{ are relatively prime. can be constructed as its perpendicular projection onto the In forward phase, we reduce the number of operations by half, thus reducing the running time of the implementation. {\displaystyle \gamma } $\phi\left(\frac{m}{a}\right)$ divides $\phi(m)$ (because $a$ and $\frac{m}{a}$ are coprime we have $\phi(a) \cdot \phi\left(\frac{m}{a}\right) = \phi(m)$), therefore we can also say that the period has length $\phi(m)$. In the same paper he also solved the problem of determining which regular polygons are constructible: The heuristics used in previous implementation works quite well in practice. Q x nm)$. x {\displaystyle \mathbb {Q} (\alpha _{1},\dots ,a_{i-1})} = a_{n1} x_1 + a_{n2} x_2 + &\dots + a_{nm} x_m = b_n [20], Pierre Wantzel(1837) proved algebraically that the problems of doubling the cube and trisecting the angle [46], The study of constructible numbers, per se, was initiated by Ren Descartes in La Gomtrie, an appendix to his book Discourse on the Method published in 1637. It follows that every algebraically constructible number is geometrically constructible, by using these techniques to translate a formula for the number into a construction for the number. [1] Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes. 0 [38] The restriction to compass and straightedge is essential to the impossibility of the classic construction problems. , Now we consider the general case, where $n$ and $m$ are not necessarily equal, and the system can be degenerate. Leibniz defined it as the line through a pair of infinitely close points on the curve. This means that when we work in the field of real numbers, the system potentially has infinitely many solutions. Then, the algorithm adds the first row to the remaining rows such that the coefficients in the first column becomes all zeros. q . :[24]. Here is an implementation using factorization in $O(\sqrt{n})$: If we need all all the totient of all numbers between $1$ and $n$, then factorizing all $n$ numbers is not efficient. [39], Although not one of the classic three construction problems, the problem of constructing regular polygons with straightedge and compass is often treated alongside them. S Given a system of $n$ linear algebraic equations (SLAE) with $m$ unknowns. 2 Implicit pivoting compares elements as if both lines were normalized, so that the maximum element would be unity. For instance the divisors of 10 are 1, 2, 5 and 10. If at least one solution exists, then it is returned in the vector $ans$. O Alternatively, they may be defined as the points in the complex plane given by algebraically constructible complex numbers. {\displaystyle a} The cosine or sine of the angle ( Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers. Thus, the solution turns into two-step: First, Gauss-Jordan algorithm is applied, and then a numerical method taking initial solution as solution in the first step. [23], Trigonometric numbers are the cosines or sines of angles that are rational multiples of [8], Equivalent definitions are that a constructible number is the In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansions[14], The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. are geometrically constructible numbers, point So, some of the variables in the process can be found to be independent. , Q However, in case the module is equal to two, we can perform Gauss-Jordan elimination much more effectively using bitwise operations and C++ bitset data types: Since we use bit compress, the implementation is not only shorter, but also 32 times faster. The function returns the number of solutions of the system $(0, 1,\textrm{or } \infty)$. It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that is a transcendental number. A {\displaystyle x} S are impossible to solve if one uses only compass and straightedge. is constructible if and only if there is a closed-form expression for -gons eluded them. ) , make the following two definitions:[5], Then, the points of Thus, the constructible numbers (defined in any of the above ways) form a field. In the reverse direction, if In many implementations, when $a_{ii} \neq 0$, you can see people still swap the $i$th row with some pivoting row, using some heuristics such as choosing the pivoting row with maximum absolute value of $a_{ji}$. You are asked to solve the system: to determine if it has no solution, exactly one solution or infinite number of solutions. The algorithm consists of $m$ phases, in each phase: So, the final complexity of the algorithm is $O(\min (n, m) . It is worth noting that the method presented in this article can also be used to solve the equation modulo any number p, i.e. This means that on the $i$th column, starting from the current line, all contains zeros. {\displaystyle (1,0)} x O x Without this heuristic, even for matrices of size about $20$, the error will be too big and can cause overflow for floating points data types of C++. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent. {\displaystyle S} y A {\displaystyle x} In a sense, it behaves as if vector $b$ was the $m+1$-th column of matrix $A$. You can check this by assigning zeros to all independent variables, calculate other variables, and then plug in to the original SLAE to check if they satisfy it. For example, if one of the equation was multiplied by $10^6$, then this equation is almost certain to be chosen as pivot in first step. x , : Strictly speaking, the method described below should be called "Gauss-Jordan", or Gauss-Jordan elimination, because it is a variation of the Gauss method, described by Jordan in 1887. In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.This theorem can be written as an equation relating the The divisor sum property also allows us to compute the totient of all numbers between 1 and $n$. [40] Gauss's treatment was algebraic rather than geometric; in fact, he did not actually construct the polygon, but rather showed that the cosine of a central angle was a constructible number. . is the length of a constructible line segment, then intersecting the Descartes associated numbers to geometrical line segments in order to display the power of his philosophical method by solving an ancient straightedge and compass construction problem put forth by Pappus. If the test solution is successful, then the function returns 1 or, Search and reshuffle the pivoting row. An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. {\displaystyle x} Angle trisection, for instance, can be done in many ways, several known to the ancient Greeks. to be the set of points that can be constructed with compass and straightedge starting with 0 Thus, using the first three properties, we can compute $\phi(n)$ through the factorization of $n$ (decomposition of $n$ into a product of its prime factors). ) r O y The last column of this matrix is vector $b$. Any equation can be replaced by a linear combination of that row (with non-zero coefficient), and some other rows (with arbitrary coefficients). is a constructible point. ( [18] Using slightly different terminology, a real number is constructible if and only if it lies in a field at the top of a finite tower of real quadratic extensions, Analogously to the real case, a complex number is constructible if and only if it lies in a field at the top of a finite tower of complex quadratic extensions. a This page was last edited on 15 August 2022, at 02:40. \end{align}$$, $$\begin{align} In geometry and algebra, a real number + b | Similarly, we perform the second step of the algorithm, where we consider the second column of second row. A such that, for each a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e., the sufficient conditions given by Gauss are also necessary). 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