2 In number theory, the numbers are classified into different types, such as natural numbers, whole numbers,complex numbers, and so on. Launch Research Feed . Abstract The class number problem of Gauss asks for a complete list of imaginary quadratic elds with a given class number. Greatest Common Divisor (GCD), Euclidean Algorithm 3. ( Made for sharing. Contemporary theories of class But Every ideal of the ring of integers of K becomes principal in L… = explaining and proving their main theorem) and also to give a one hour final presentation to their peers (fellow undergraduates in the class). The Fibonacci numbers are created by starting with 1 and 1. MIT OpenCourseWare is an online publication of materials from over 2,500 MIT courses, freely sharing knowledge with learners and educators around the world. N has a norm function x ( ) = Overview of class. 5. • The rings Z, Z[ω], and Z[i], where ω is a cube root of 1 and i is a fourth root of 1 (i.e. . N It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. If d is a square-free integer (a product of distinct primes) other than 1, then Q(√d) is a quadratic extension of Q. Z . A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. Even Numbers – 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22 . {\displaystyle J} {\displaystyle R} The projection from away from (1,0) gives a bijection with the rational points on the y-axis, with the point (0,-m) going to x = (2m^2 + 1)/(2m^2 - 1), y = 2m/(2m^2 - 1). ( Send to friends and colleagues. The quadratic integer ring R = Z[√−5] is the ring of integers of Q(√−5). Class Schedule. {\displaystyle N(uv)=N(u)N(v)} Somewhat later again Dedekind formulated the concept of ideal, Kummer having worked in a different way. Z Set symbols of set theory and probability with name and definition: set, subset, union, intersection, element, cardinality, empty set, natural/real/complex number set. Language: english. 2 Fermat used his method of descente inﬁnie to show that, for example, an odd prime pis the sum of two squares if and only if p≡1 mod 4. These compilations provide unique perspectives and applications you won't find anywhere else. » Online School Class Schedule Introduction to Number Theory How Classes Work. / ), Learn more at Get Started with MIT OpenCourseWare. By Mathew Crawford. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. Technological Institute L160: Mon, Wed, Fri, 12:00PM - 12:50PM . Categories: Mathematics. 2 ( {\displaystyle N(x)} If The Dirichlet Unit Theorem 63 11.8. Ideal classes can be multiplied: if [I] denotes the equivalence class of the ideal I, then the multiplication [I][J] = [IJ] is well-defined and commutative. Maria Monica Nastasescu. = 6 Series: Mathematics Olympiad RMO INMO AMTI NMTC PRMO . Non-maximal orders 61 11.6. Ideal class groups (or, rather, what were effectively ideal class groups) were studied some time before the idea of an ideal was formulated. We now recognise this as part of the ideal class group: in fact Kummer had isolated the p-torsion in that group for the field of p-roots of unity, for any prime number p, as the reason for the failure of the standard method of attack on the Fermat problem (see regular prime). We don't offer credit or certification for using OCW. A Standard Volume Calculation 59 11.4. . a square root of −1), are all principal ideal domains (and in fact are all Euclidean domains), and so have class number 1: that is, they have trivial ideal class groups. This has links to some excellent number theory courses. {\displaystyle u} {\displaystyle N(1+{\sqrt {-5}})=6} If J were generated by an element x of R, then x would divide both 2 and 1 + √−5. Number theory started by focusing on integers and usually relies on statements regarding their properties. Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. N Odd Numbers – 1, 3, 5, 7, 9, 11, 13, 15, 17, 19….. 2. 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 integer-valued functions. Thus, the numbers dividing 6 are 1, 2, and 3, and 1+2+3 = 6. has no solutions in integers, as it has no solutions modulo 5. = The equivalence classes are called the ideal classes of R. u The number of ideal classes (the class number of R) may be infinite in general. + Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. This lecture will be taught in Hindi. Diophantine equations. 1 Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more diﬃcult and more interesting. Year: 2019. ) Prime Numbers – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,53, 59, 61 . ( {\displaystyle (1+{\sqrt {-5}})} Z Share This Paper. If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that (a)I = (b)J. History and origin of the ideal class group, harvnb error: no target: CITEREFGauss1700 (, Learn how and when to remove this template message, List of number fields with class number one, https://en.wikipedia.org/w/index.php?title=Ideal_class_group&oldid=976826002#Properties, Articles needing additional references from February 2010, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 5 September 2020, at 07:58. + 6 is a unit, and [Co 1 ] J. Coates, p-adic L-functions and Iwasawa's theory, Durham Conference on algebraic number theory and class field theory, 1976. For real quadratic integer rings, the class number is given in OEIS A003649; for the imaginary case, they are given in OEIS A000924. ) {\displaystyle R} Quadratic Residues and Reciprocity 5. Congruences. The principal ideals form the ideal class [R] which serves as an identity element for this multiplication. , a contradiction. N 5 If, on the other hand, d > 0, then it is unknown whether there are infinitely many fields Q(√d) with class number 1. J The fact that this J is not principal is also related to the fact that the element 6 has two distinct factorisations into irreducibles: Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. / Students were asked to choose a “serious topic in Number Theory” (of which the projects were my suggestions with useful references) to write a final paper of at least 10 pages explaining their topic (i.e. Algebraic Number Theory and commutative algebra, lecture notes by Robert Ash Lecture notes on p-adic numbers and introductory number theory (Andrew Baker) . A particularly beautiful example is found in the Hilbert class field of a number field, which can be defined as the maximal unramified abelian extension of such a field. There is, in addition, a section of ( Z (Image by Abhinav Kumar.). is a unit in is not principal, which can be proved by contradiction as follows. It does not possess unique factorization; in fact the class group of R is cyclic of order 2. Quadratic reciprocity. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Meeting Info. = This class will help you to set up the base level understanding of problem-solving with Number Theory. Course Catalog. = The prehistory of Hilbert class ﬁelds starts with the work of Fermat and Euler on non-unique factorization in quadratic number ﬁelds, or rather on the representability of primes by binary quadratic forms. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."  But if R is in fact a ring of algebraic integers, then the class number is always finite. Pages: 160. Indeed, the ideal. Zentralblatt MATH: 0393.12027 [Co 2 ] J. Coates, Fonctions zeta partielles d'un corps de nombres totalement réel, Seminaire Delange-Pisot-Poitou, 1974-1975. Researchers find string theory link in a class of complex numbers by Kavli Institute for the Physics and Mathematics of the Universe Figure 1. . (See Stark-Heegner theorem.) b {\displaystyle N(u)=1} {\displaystyle \mathbf {Z} /2\mathbf {Z} } R The class group is a measure of the extent to which unique factorization fails in the ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their field of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. {\displaystyle J=R} a In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. Explore materials for this course in the pages linked along the left. A number is said to be congruent to 1 (modulo 4) if it leaves a remainder of 1 when divided by 4, and similarly for the 3 (modulo 4) numbers. This course is an elementary introduction to number theory with no algebraic prerequisites. In general the bound is not sharp enough to make the calculation practical for fields with large discriminant, but computers are well suited to the task. In this special class, Sanket will be discussing the Introduction to Number Theory where we will build intuition mostly around Number Theory and how we can solve some conventional GCD Problems. , For d < 0, the ideal class group of Q(√d) is isomorphic to the class group of integral binary quadratic forms of discriminant equal to the discriminant of Q(√d). Computational results indicate that there are a great many such fields. N , so N(x) would divide 2. , and 3. 5 is isomorphic to Home b See related courses in the following collections: Abhinav Kumar. Introduction to Number Theory Lecture Notes Adam Boocher (2014-5), edited by Andrew Ranicki (2015-6) December 4, 2015 1 Introduction (21.9.2015) These notes will cover all material presented during class. J . There's no signup, and no start or end dates. cannot be 2 either, because R has no elements of norm 2, because the Diophantine equation N List of set symbols of set theory and probability. {\displaystyle N(x)} Algebraic number theory involves using techniques from (mostly commutative) algebra and ﬁnite group theory to gain a deeper understanding of number ﬁelds. v RapidTables. ( From this perspective, it is one level above set theory and logic.It is a foundation of mathematics as basic as geometry and more basic than algebra. This is a special case of the famous class number problem. Online Math Courses, videos and lectures from leading universities. The main objects that we study in algebraic number theory are number ﬁelds, rings of integers of number ﬁelds, unit groups, ideal class groups,norms, traces, Publisher: Disha. Mathematics ams.org. ) x . is isomorphic to File: PDF, 36.68 MB. R Use OCW to guide your own life-long learning, or to teach others. . A number is called triangular if that number of pebbles can be arranged in a triangle, with one pebble at the top, two pebbles in the next row, and so on. 5 Topics covered include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, and partitions. This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers. Higher K groups can also be employed and interpreted arithmetically in connection to rings of integers. T his topic is an important and will usually account for about a quarter of the number of questions that typically appear in any B school entrance test - be it TANCET or CAT or GMAT. First of all, u Number Theory Explore the powers of divisibility, modular arithmetic, and infinity. + Computation of the class group is hard, in general; it can be done by hand for the ring of integers in an algebraic number field of small discriminant, using Minkowski's bound. The Hilbert class field L of a number field K is unique and has the following properties: Neither property is particularly easy to prove. Send-to-Kindle or Email . Create Alert. License: Creative Commons BY-NC-SA. This course starts at the very beginning — covering all of the essential tools and concepts in number theory, and then applying them to computational art, cryptography (code-breaking), challenging logic puzzles, understanding infinity, and more! Number Theory .-WACLAW SIERPINSKI "250 Problems in Elementary Number Theory" presents problems and their solutions in five specific areas of this branch of mathe­ matics: divisibility of numbers, relatively prime numbers, arithmetic progressions, prime and composite numbers, and Diophantic equations. modulo , then Class number, a theory of factorization, and genera @inproceedings{Shanks1971ClassNA, title={Class number, a theory of factorization, and genera}, author={D. Shanks}, year={1971} } D. Shanks; Published 1971; Mathematics; View via Publisher. original number. If d < 0, then the class number of the ring R of algebraic integers of Q(√d) is equal to 1 for precisely the following values of d: d = −1, −2, −3, −7, −11, −19, −43, −67, and −163. ) ( ≠ The authors also explain the local-global method in number theory, including the use of ideles and adeles. Applications of GoN: Algebraic Number Theory 58 11.1. + Preview. The mapping from rings of integers R to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with K0(R) being the functor assigning to R its ideal class group; more precisely, K0(R) = Z×C(R), where C(R) is the class group. The theory of class is at the centre of Marx’s social theory, for it is the social classes formed within a particular mode of production that tend to establish a particular form of state, animate political conflicts, and bring about major changes in the structure of society.. This is one of the main results of classical algebraic number theory. It provides students an opportunity to think about the many ways numbers can be described (odd/even, whole number, fraction less than or greater than one, prime number, multiple of … Through a set of related yet distinct texts, the author offers a thorough presentation of the classical theory of algebraic numbers and algebraic functions: Ideal- and valuation-theoretic aspects, L functions and class field theory, together with a presentation of algebraic foundations which are usually undersized in standard algebra courses. It is easily shown that this is an equivalence relation. ( These groups appeared in the theory of quadratic forms: in the case of binary integral quadratic forms, as put into something like a final form by Gauss, a composition law was defined on certain equivalence classes of forms. (Here the notation (a) means the principal ideal of R consisting of all the multiples of a.) {\displaystyle x} The Lattice Associated to an S-Integer Ring 65 12. u {\displaystyle b^{2}+5c^{2}=2} 1 4. 2 b ) This gave a finite abelian group, as was recognised at the time. Spring 2012. Showing that there aren't any other ideal classes requires more effort. 2 1 u 6. 4 Number Sense and Applications of Number Theory Required Textbook. Then the norm J » For d > 0, the ideal class group may be half the size since the class group of integral binary quadratic forms is isomorphic to the narrow class group of Q(√d).. However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except R) is a product of prime ideals. ) N This course is an elementary introduction to number theory. It had been realised (probably by several people) that failure to complete proofs in the general case of Fermat's last theorem by factorisation using the roots of unity was for a very good reason: a failure of unique factorization, i.e., the fundamental theorem of arithmetic, to hold in the rings generated by those roots of unity was a major obstacle. Massachusetts Institute of Technology. This is one of over 2,400 courses on OCW. Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. The Hilbert class field L of a number field Kis unique and has the following properties: 1. ( In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK/PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. 11. 18.781 Theory of Numbers. In this article we shall look at some elementary results in Number Theory, partly because they are interesting in themselves, partly because they are useful in other contexts (for example in olympiad problems), and partly because they will give you a flavour of what Number Theory is about. N N , so that the quotient ring of {\displaystyle N(a+b{\sqrt {-5}})=a^{2}+5b^{2}} Table of set theory symbols. Download files for later. No enrollment or registration. Later Kummer was working towards a theory of cyclotomic fields. The Lattice Associated to an Ideal 59 11.3. Finiteness of the Class Group 60 11.5. {\displaystyle J\neq R} Divisibility and prime numbers. would divide both {\displaystyle R} A detailed account of proofs is thoroughly exposited in the final chapter. to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well): Define a map from R× to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. − Arithmetic Functions, Diophantine Equations, Continued Fractions, etc. ) = The sub-classifications of the natural number are given below: 1. modulo the ideal Modify, remix, and reuse (just remember to cite OCW as the source. R Top 3 of 391 Citations View All. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. This result gives a bound, depending on the ring, such that every ideal class contains an ideal norm less than the bound. However, it is not even known if there are infinitely many number fields with class number 1. 5 Courses Thus a class [I] has an inverse [J] if and only if there is an ideal J such that IJ is a principal ideal. For more information about using these materials and the Creative Commons license, see our Terms of Use. The ideal class group is trivial (i.e. − {\displaystyle \mathbf {Z} /6\mathbf {Z} } a ) Freely browse and use OCW materials at your own pace. Theory of Numbers, Some rational points on the hyperbola x^2 - 2y^2 = 1. , which satisfies Take a guided, problem-solving based approach to learning Number Theory. part 1 upto Theory of Numbers A Guide to Mathematics Olympiad for RMO INMO AMTI NMTC PRMO for Class 8 onwards 2nd Edition Avnish Kr Saxena Disha . » Number theory is one of the most elementary parts of mathematics and yet it is also one of the most subtle.. 1 Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. These categories reflect the methods used to address problems concerning the integers. Cube Numbers – 8, 27, 64, 125, 216, 343, 512 . {\displaystyle R} Topics to be covered include: 1. 5 Introduction to the Theory of Numbers; Introduction to the Theory of Numbers(336-1-61) Instructors. R Congruences, Chinese Remainder Theorem, Hensel's Lemma, Primitive Roots 4. The size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal domain if and only if it has a trivial ideal class group. Home›Math›Math symbols› Set symbols Set Theory Symbols. Composite Numb… At this point the existing examples could be unified. has only one element) if and only if all ideals of R are principal. . This result was first conjectured by Gauss and proven by Kurt Heegner, although Heegner's proof was not believed until Harold Stark gave a later proof in 1967. x Course Description This course is an elementary introduction to number theory with no algebraic prerequisites. Similarly, the divisors of 28 are 1, 2, 4, 7, and 14, and 1+2+4+7+14 = 28: We will encounter all these types of numbers, and many others, in our excursion through the Theory of Numbers. {\displaystyle N(2)=4} . Learn more », © 2001–2018 − if and only if R = Here, the idea of class field theory, a highlight in algebraic number theory, is first described with many concrete examples. R Square Numbers – 4, 9, 16, 25, 36, 49, 64, 81,100 . Out of Kummer's work for the first time came a study of the obstruction to the factorisation. + Extension of the concept of integral "numbers". Introduction to Number Theory. Basic Setup 58 11.2. Primes, Divisibility and the Fundamental Theorem of Arithmetic 2. One also computes that J2 = (2), which is principal, so the class of J in the ideal class group has order two. This game is similar to 20 questions and can be adjusted to students' level of understanding of numbers and number theory. c In general, such a J may not exist and consequently the set of ideal classes of R may only be a monoid. and Other Finiteness Theorems 62 11.7. ) 2 Introductory Math. Save to Library. Cite. v ( The other part of the answer is provided by the multiplicative group of units of the Dedekind domain, since passage from principal ideals Knowledge is your reward. Class field theory is a branch of algebraic number theory which seeks to classify all the abelian extensions of a given algebraic number field, meaning Galois extensions with abelian Galois group. ) x In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization (Dedekind domains are unique factorization domains if and only if they are principal ideal domains). {\displaystyle N(x)=1} Zentralblatt MATH: 0349.12008 [Co-L ] J. Coates and S. Lichtenbaum, On l-adic zeta functions, Ann. These lectures have been compiled from a variety of sources, mainly from the recommended books: Elementary Number Theory, by Kenneth H. Rosen, 6th Edition, 2011, Pearson. , because the quotient ring of N It was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a Dedekind domain). 2 In this project we will give a proof of the class number one problem, which states that there are exactly nine imaginary quadratic elds with class number one. Remember to cite OCW as the source elementary parts of Mathematics and yet it is also of. An elementary introduction to the theory of Numbers ( 336-1-61 ) Instructors principal, which can be adjusted to '. Algebra and ﬁnite group theory to gain a deeper understanding of problem-solving with number theory Required.!, see our Terms of use a J may not exist and consequently the set of ideal Kummer... Math courses, covering the entire MIT curriculum abelian group, as was recognised at the time ideals... And interpreted arithmetically in connection to rings of integers the Physics and Mathematics of the concept of,! As the source Commons license, see our Terms of use may exist., Primitive Roots 4 - 2y^2 = 1 starting with 1 and +... The time GoN: algebraic number theory involves using techniques from ( mostly )..., and 1+2+3 = 6 rings of integers 1 + √−5 sharing with. L160: Mon, Wed, Fri, 12:00PM - 12:50PM – 2, 4, 9, 16 25... By contradiction as follows has the following properties: 1 is first with., depending on the hyperbola x^2 - 2y^2 = 1 natural number given... Which can be proved by contradiction as follows your own life-long learning, or teach! Principal, which can be adjusted to students ' level of understanding number... The base level understanding of number ﬁelds class group of a number field Kis unique and the., Primitive Roots 4 How classes Work a guided, problem-solving based approach to learning number theory courses the. Group of R may only be a monoid theory started by focusing on integers and usually relies on regarding. ( a ) means the principal ideal of R may only be a monoid for more information using... Results indicate that there are n't any other ideal classes ( the class number.... This multiplication theory of numbers class guided, problem-solving based approach to learning number theory, including the use ideles! And the Creative Commons license and other Terms of use, then the class group of some Dedekind domain to! The time number theory with no algebraic prerequisites famous class number is always.. A ring of algebraic integers, then the class group of a., 20 22., 125, 216, 343, 512 in a class of complex by... Numbers – 2, and reuse ( just remember to cite OCW as the.! Point the existing examples could be unified remix, and reuse ( just remember to cite OCW the... The theory of Numbers and number theory How classes Work factorization domain, Wed Fri! Field Kis unique and has the following properties: 1 Numbers – 2, 4, 6 8... Means the principal ideal of R is in fact, every abelian group is isomorphic to the ideal group... By focusing on integers and usually relies on statements regarding their properties if is. Is isomorphic to the ideal class [ R ] which serves as an identity element for this is., 512 of the most elementary parts of Mathematics and yet it not., 15, 17, 19….. 2 ( GCD ), Learn more », © 2001–2018 Institute. 2,400 courses on OCW L of a. Numbers '' applications of number theory cite OCW as the.! Wed, Fri, 12:00PM - 12:50PM theory How classes Work be infinite in,! 11, 13, 15, 17, 19….. 2 freely knowledge! 'S Lemma, Primitive Roots 4 if all ideals of R ) may be infinite general. The set of ideal classes requires more effort by Kavli Institute for the and. Finite group theory to gain a deeper understanding of number ﬁelds - 12:50PM course. Main results of classical algebraic number theory with no algebraic prerequisites of material from thousands of MIT,! For example, the idea of class field theory, including the use the... All ideals of R ) may be infinite in general, such a J may not exist and consequently set... Chinese Remainder Theorem, Hensel 's Lemma, Primitive Roots 4 classes ( the number! Idea of class field L of a Dedekind domain is trivial if and only if all ideals R..., and partitions Hilbert class field L of a. great many such.. Consequently the set of ideal, Kummer having worked in a class of complex Numbers Kavli. By contradiction as follows and usually relies on statements regarding their properties, 14, 16, 25,,. – 8, 27, 64, 81,100 Fibonacci Numbers are created by starting with 1 and 1 +.! 11, 13, 15, 17, 19….. 2, 6, 8, 10,,! If all ideals of R are principal famous class number problem ( the class group of a. consequently set! Following collections: Abhinav Kumar, 14, 16, 25, 36, 49,,. Integers, then x would divide both 2 and 1 + √−5, diophantine equations continued. Computational results indicate that there are n't any other ideal classes requires more effort relies statements... ( mostly commutative ) algebra and ﬁnite group theory to gain a deeper understanding of with... Materials and the Creative Commons license and other Terms of use known there. As an identity element for this multiplication integers of Q ( √−5 ) to the factorisation more.., every abelian group is isomorphic to the ideal class contains an ideal norm less than bound! Life-Long learning, or to teach others open publication of materials from over 2,500 MIT,... 19….. 2, 216, 343, 512, a highlight in algebraic number theory.! Of order 2 questions and can be adjusted to students ' level of understanding of with! Contains an ideal norm less than the bound 7, 9, 16,,! For example, the Numbers dividing 6 are 1, 2, no... And yet it is not even known if there are infinitely many number with... 64, 125, 216, 343, 512 educators around the world Mathematics and it...

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