The Markov numbers are the union of the solutions to the Markov equation(1)and are related to Lagrange numbers by(2)The first few solutions are , (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), .... All solutions can be generated from the first two of these since the equation is a quadratic in each of the variables, so one integer solution leads to a second, and it turns out that all solutions (other than the first two singular ones) have distinct values of , , and , and share two of their three values with three other solutions (Guy 1994, p. 166). The Markov numbers are then given by 1, 2, 5, 13, 29, 34, ... (OEIS A002559).The Markov numbers for triples in which one term is 5 are 1, 2, 13, 29, 194, 433, ... (OEIS A030452), whose terms are given by the recurrence relation(3)with , , , and .The solutions can be arranged in an infinite tree with two smaller branches on each trunk. It is not known if two different regions can have the same label. Strangely, the regions..
A set of distinct positive integers satisfies the Diophantus property of order (a positive integer) if, for all , ..., with ,(1)the s are integers. The set is called a Diophantine -tuple.Diophantine 1-doubles are abundant: (1, 3), (2, 4), (3, 5), (4, 6), (5, 7), (1, 8), (3, 8), (6, 8), (7, 9), (8, 10), (9, 11), ... (OEIS A050269 and A050270). Diophantine 1-triples are less abundant: (1, 3, 8), (2, 4, 12), (1, 8, 15), (3, 5, 16), (4, 6, 20), ... (OEIS A050273, A050274, and A050275).Fermat found the smallest Diophantine 1-quadruple: (Davenport and Baker 1969, Jones 1976). There are no others with largest term , and Davenport and Baker (1969) showed that if , , and are all squares, then .General quadruples are(2)where are Fibonacci numbers, and(3)The quadruplet(4)is (Dujella 1996). Dujella (1993) showed there exist no Diophantine quadruples .A longstanding conjecture is that no integer Diophantine quintuple exists (Gardner 1967, van Lint 1968, Davenport..
Find nontrivial solutions to other than , where is the divisor function. Nontrivial solutions means that solutions which are multiples of smaller solutions are not considered. For example, multiples of are solutions for , 7, 9, 11, 13, 17, 19, 23, 21, ....Nontrivial solutions to Wallis's equation include , (326, 407), (406, 489), (627, 749), (740, 878), (880, 1451), (888, 1102), (1026, 1208), (1110, 1943), (1284, 1528, 1605), (1510, 1809), (1628, 1630, 2035), (1956, 2030, 2445), (2013, 2557), (2072, 3097), (2508, 2996, 3135, 3745), ....
The Diophantine equationThe Markov numbers are the union of the solutions to this equation and are related to Lagrange numbers.
A set of positive integers is said to be Diophantine iff there exists a polynomial with integral coefficients in indeterminates such thatIt has been proved that the set of prime numbersis a Diophantine set.
The 2-1 equation(1)is a special case of Fermat's last theorem and so has no solutions for . Lander et al. (1967) give a table showing the smallest for which a solution to(2)with is known. An updated table is given below; a more extensive table may be found at Meyrignac's web site.1234562233243254367537765488755910986510131211976Take the results from the Ramanujan 6-10-8 identity that for , with(3)and(4)then(5)Using(6)(7)now gives(8)for or 4.
A Thue equation is a Diophantine equation of the formin terms of an irreducible polynomial of degree having coefficients for which solutions in integers and are sought for each given constant with .Thue (1909) proved that such an equation has only finitely many solutions, but it was not until much later that Tzanakis and de Weger (1989) gave a practical algorithm for finding bounds on and . Although these bounds can be astronomically large in some cases, they are typically small enough to allow an exhaustive search for all solutions.
The 10.1.2 equation(1)is a special case of Fermat's last theorem with , and so has no solution. No solutions are known with . A 10.1.13 solution is(2)(S. Chase). The smallest 10.1.15 solution is(3)(J.-C. Meyrignac 1999). The smallest 10.1.22 solution is(4)(Ekl 1998). The smallest 10.1.23 solution is(5)(Lander et al. 1967).10.2.12 solutions include (6)(7)(V. Pliousnine 2000, N. Kuosa 2000). The smallest 10.2.13 solution is(8)The smallest 10.2.15 solution is(9)(Ekl 1998). The smallest 10.2.19 solution is(10)(Lander et al. 1967). A 10.3.11 solution is(11)(J. Wroblewski 2002). A 10.3.12 solution is(12)(T. Nolan 2000). The smallest 10.3.13 solution is(13)The smallest 10.3.14 solution is(14)(Ekl 1998). The smallest 10.3.24 solution is(15)(Lander et al. 1967).A 10.4.9 solution is(16)(J. Wroblewski 2002). 10.4.10 solutions include (17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(J. Wroblewski..
The 9.1.2 equation(1)is a special case of Fermat's last theorem with , and so has no solution. No 9.1.3, 9.1.4, 9.1.5, 9.1.6, 9.1.7, 9.1.8, or 9.1.9 solutions are known. A 9.1.10 solution is(2)(J. Wroblewski 2002), and two 9.1.11 solutions are given by (3)(4)(5)(6)(S. Chase; Aloril 2002). The smallest 9.1.12 solution is(7)(Meyrignac 1997). No 9.1.13 solution is known. The smallest 9.1.14 solution is(8)(Ekl 1998).No 9.2.2, 9.2.3, 9.2.4,. 9.2.5, 9.2.6, 9.2.7, or 9.2.8 solutions are known. 9.2.9 solutions include (9)(10)(11)(12)(J. Wroblewski 2002). A 9.2.10 solution is given by(13)(L. Morelli 1999). No 9.2.11 solutions are known. The smallest 9.2.12 solution is(14)(Lander et al. 1967, Ekl 1998). There are no known 9.2.13 or 9.2.14 solutions.The smallest 9.2.15 solution is(15)(Lander et al. 1967).There are no known 9.3.3, 9.3.4, 9.3.5, 9.3.6, 9.3.7, or 9.3.8 solutions. The smallest 9.3.9 solution is(16)(Ekl..
The 8.1.2 equation(1)is a special case of Fermat's last theorem with , and so has no solution. No 8.1.3, 8.1.4, 8.1.5, 8.1.6, or 8.1.7 solutions are known. The only known 8.1.8 is(2)(S. Chase; Meyrignac). The smallest 8.1.9 is(3)(N. Kuosa). The smallest 8.1.10 is(4)(N. Kuosa, PowerSum). The smallest 8.1.11 solution is(5)(Lander et al. 1967, Ekl 1998). The smallest 8.1.12 solution is(6)(Lander et al. 1967). The general identity(7)gives a solution to the 8.1.17 equation (Lander et al. 1967).No 8.2.2, 8.2.3, 8.2.4, 8.2.5, or 8.2.6 solution is known. A single 8.2.7 solutions is known,(8)(S. Chase; Meyrignac). The smallest 8.2.8 solution is(9)The smallest 8.2.9 solution is(10)(Lander et al. 1967, Ekl 1998).No 8.3.3 or 8.3.4 solutions are known. An 8.3.5 solution is(11)(S. Chase, Meyrignac, Resta and Meyrignac 2003). No 8.3.6 solution is known. The smallest 8.3.7 solution is(12)The smallest 8.3.8 solution..
The 7.1.2 equation(1)is a special case of Fermat's last theorem with , and so has no solution. No solutions to the 7.1.3, 7.1.4, 7.1.5, 7.1.6 equations are known. There is now a known solutions to the 7.1.7 equation,(2)(M. Dodrill 1999, PowerSum), requiring an update by Guy (1994, p. 140). The smallest 7.1.8 solution is(3)(Lander et al. 1967, Ekl 1998). The smallest 7.1.9 solution is(4)(Lander et al. 1967).No solutions to the 7.2.2, 7.2.3, 7.2.4, or 7.2.5 equations are known. The smallest 7.2.6 equation is(5)(Meyrignac). The smallest 7.2.8 solution is(6)(Lander et al. 1967, Ekl 1998). A 126.96.36.199 solution is(7)(8)(Lander et al. 1967).No solutions to the 7.3.3 equation are known (Ekl 1996), nor are any to 7.3.4. The smallest 7.3.5 equations are(9)(10)No solutions are known to the 7.3.6 equation. The smallest 7.3.7 solution is(11)(Lander et al. 1967).Guy (1994, p. 140) asked if a 7.4.4 equation exists. The following..
The 6.1.2 equation(1)is a special case of Fermat's last theorem with , and so has no solution. No 6.1. solutions are known for (Lander et al. 1967; Guy 1994, p. 140). The smallest 6.1.7 solution is(2)(Lander et al. 1967; Ekl 1998). The smallest primitive 6.1.8 solutions are(3)(Lander et al. 1967). The smallest 6.1.9 solution is(4)(Lander et al. 1967). The smallest 6.1.10 solution is(5)(Lander et al. 1967). The smallest 6.1.11 solution is(6)(Lander et al. 1967). There is also at least one 6.1.16 identity,(7)(Martin 1893). Moessner (1959) gave solutions for 6.1.16, 6.1.18, 6.1.20, and 6.1.23 equations.Ekl (1996) has searched and found no solutions to the 6.2.2(8)with sums less than . No solutions are known to the 6.2.3 or 6.2.4 equations. The smallest primitive 6.2.5 equations are(9)(10)(11)(12)(13)(E. Brisse 1999, Resta 1999, Resta and Meyrignac 2003, Meyrignac). The smallest 6.2.6 equation is(14)(Ekl 1998). The smallest..
The 5.1.2 fifth-order Diophantine equation(1)is a special case of Fermat's last theorem with , and so has no solution. improving on the results on Lander et al. (1967), who checked up to . (In fact, no solutions are known for powers of 6 or 7 either.) No solutions to the 5.1.3 equation(2)are known (Lander et al. 1967). For 4 fifth powers,the 5.1.4 equation has solutions(3)(4)(Lander and Parkin 1967, Lander et al. 1967, Ekl 1998), the second of which was found by J. Frye (J.-C. Meyrignac, pers. comm., Sep. 9, 2004), but it is not known if there is a parametric solution (Guy 1994, p. 140). Sastry (1934) found a 2-parameter solution for 5.1.5 equations(5)(quoted in Lander and Parkin 1967), and Lander and Parkin (1967) found the smallest numerical solutions. Lander et al. (1967) give a list of the smallest solutions, the first few being(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(Lander and Parkin 1967, Lander et al...
In 1913, Ramanujan asked if the Diophantineequation of second ordersometimes called the Ramanujan-Nagell equation, has any solutions other than , 4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; OEIS A060728). These correspond to , 3, 5, 11, and 181 (OEIS A038198). Nagell (1948) and Skolem et al. (1959) showed there are no solutions past , thus establishing Ramanujan's question in the negative.A generalization to two variables and was considered by Euler (Engel 1998, p. 126).
As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exist a general algorithm for solving a general quartic Diophantine equation. However, the algorithm for constructing such an unsolvable quartic Diophantine equation can require arbitrarily many variables (Matiyasevich 1993).As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 19 positive biquadrates (), that every "sufficiently large" integer is a sum of no more than 16 positive biquadrates (), and that every integer is a sum of at most 10 signed biquadrates (; although it is not known if 10 can be reduced to 9). The first few numbers which are a sum of four fourth powers ( equations) are 353, 651, 2487, 2501, 2829, ... (OEIS A003294).The 4.1.2 equation(1)is a case of Fermat's last theorem with and therefore has no solutions. In fact, the equations(2)also have no solutions..
A triple of positive integers satisfying is said to be harmonic ifIn particular, such a triple is harmonic if the reciprocals of its terms form an arithmetic sequence with common difference whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of harmonic triples and the set of equivalence classes of geometric triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .
As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (), that every "sufficiently large" integer is a sum of no more than 7 positive cubes (; although it is not known if 7 can be reduced), and that every integer is a sum of at most 5 signed cubes (; although it is not known if 5 can be reduced to 4).It is known that every can be written in the form(1)An elliptic curve of the form for an integer is known as a Mordell curve.The 3.1.2 equation(2)is a case of Fermat's last theorem with . In fact, this particular case was known not to have any solutions long before the general validity of Fermat's last theorem was established. Thue showed that a Diophantine equation of the form(3)for , , and integers, has only finite many solutions (Hardy 1999, pp. 78-79).Miller and Woollett (1955) and Gardiner et al. (1964) investigated integersolutions of(4)i.e., numbers representable as the..
A general quadratic Diophantine equation in two variables and is given by(1)where , , and are specified (positive or negative) integers and and are unknown integers satisfying the equation whose values are sought. The slightly more general second-order equation(2)is one of the principal topics in Gauss's Disquisitiones arithmeticae. According to Itô (1987), equation (2) can be solved completely using solutions to the Pell equation. In particular, all solutions of(3)are among the convergents of the continued fractions of the roots of .Solution to the general bivariate quadratic Diophantine equation is implemented in the Wolfram Language as Reduce[eqn && Element[x|y, Integers], x, y].For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. L. Siegel.An equation of the form(4)where is an integer is a very special type of equation called a Pell equation...
A Diophantine equation is an equation in which only integersolutions are allowed.Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution. Such an algorithm does exist for the solution of first-order Diophantine equations. However, the impossibility of obtaining a general solution was proven by Yuri Matiyasevich in 1970 (Matiyasevich 1970, Davis 1973, Davis and Hersh 1973, Davis 1982, Matiyasevich 1993) by showing that the relation (where is the th Fibonacci number) is Diophantine. More specifically, Matiyasevich showed that there is a polynomial in , , and a number of other variables , , , ... having the property that iff there exist integers , , , ... such that .Matiyasevich's result filled a crucial gap in previous work by Martin Davis, Hilary Putnam, and Julia Robinson. Subsequent work by Matiyasevich and Robinson proved that even for equations in thirteen variables,..
A generalization of the equation whose solution is desired in Fermat'slast theoremtofor , , , and positive constants, with trivial solutions having , , or being excluded. is trivial to solve by taking and . is more difficult, but can be solved by noting that solutions exist for values of which can be written as a sum of two squares, the first few of which are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, ... (OEIS A001481).
A Pythagorean quadruple is a set of positive integers , , , and that satisfy(1)For positive even and , there exist such integers and ; for positive odd and , no such integers exist (Oliverio 1996).Examples of primitive Pythagorean quadruples include , , , , , and .Oliverio (1996) gives the following generalization of this result. Let , where are integers, and let be the number of odd integers in . Then iff (mod 4), there exist integers and such that(2)A set of Pythagorean quadruples is given by(3)(4)(5)(6)where , , and are integers (Mordell 1969). This does not, however, generate all solutions. For instance, it excludes (36, 8, 3, 37).
Find a square number such that, when a given integer is added or subtracted, new square numbers are obtained so that(1)and(2)This problem was posed by the mathematicians Théodore and Jean de Palerma in a mathematical tournament organized by Frederick II in Pisa in 1225. The solution (Ore 1988, pp. 188-191) is(3)(4)where and are integers. and are then given by(5)(6)Fibonacci proved that all numbers (the congrua) are divisible by 24. Fermat's right triangle theorem is equivalent to the result that a congruum cannot be a square number.A table for small and is given in Ore (1988, p. 191), and a larger one (for ) by Lagrange (1977). The firstSloaneA057103A055096A057104A057105212457131961014232120131774124017237423842028443336253117
A number which satisfies the conditions of the congruum problem:andwhere are integers. The list of congrua is given by 24, 96, 120, 240, 336, 384, 480, 720, ... (OEIS A057102).
A congruent number can be defined as an integer that is equal to the area of a rational right triangle (Koblitz 1993).Numbers such that(1)are also known as congruent numbers. They are a generalization of the congruum problem, which is the case .For example, , the smallest congruent numbers are(2)(3)(4)(5)
Consider solutions to the equation(1)Real solutions are given by for , together with the solution of(2)which is given by(3)where is the Lambert W-function. This function is illustrated above by the blue curve.Rational parametric solutions are given by(4)(5)for , , ... (Dunn 1980, Pickover 2002). These solutions are shown on the plot as red dots.
A concordant form is an integer triple where(1)with and integers. Examples include(2)Dickson (2005) states that C. H. Brooks and S. Watson found in The Ladies' and Gentlemen's Diary (1857) that and can be simultaneously squares for only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 99, and 100 (which evidently omits 47, 53, and 83 from above). The list of concordant primes less than 1000 is now complete with the possible exception of the 16 primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593, 607, 641, 743, 821, 929, and 971.
An integer which is expressible in more than one way in the form or where is relatively prime to . If the integer is expressible in only one way, it is called a monomorph.
Find consecutive powers, i.e., solutions toexcluding 0 and 1. Catalan's conjecture states that the only solution is , so 8 and 9 ( and ) are the only consecutive powers (again excluding 0 and 1).
For every , there exist only finite many pairs of powers with and natural numbers and .
Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation has no integer solutions for and .The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number..
The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1). In other words,(1)is the only nontrivial solution to Catalan'sDiophantine problem(2)The special case and is the case of a Mordell curve.Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000).This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in..
The Diophantine equationThe assertion that this equation has no nontrivial solutions for has a long and fascinating history and is known as Fermat's last theorem.
Solve the Pell equationin integers. The smallest solution is , .
A generalization of Fermat's last theorem which states that if , where , , , , , and are any positive integers with , then , , and have a common factor. The conjecture was announced in Mauldin (1997), and a cash prize of has been offered for its proof or a counterexample (Castelvecchi 2013).This conjecture is more properly known as the Tijdeman-Zagier conjecture (Elkies 2007).
A special case of the quadratic Diophantineequation having the form(1)where is a nonsquare natural number (Dickson 2005). The equation(2)arising in the computation of fundamental units is sometimes also called the Pell equation (Dörrie 1965, Itô 1987), and Dörrie calls the positive form of (2) the Fermat difference equation. While Fermat deserves credit for being the first to extensively study the equation, the erroneous attribution to Pell was perpetrated by none other than Euler himself (Nagell 1951, p. 197; Burton 1989; Dickson 2005, p. 341). The Pell equation was also solved by the Indian mathematician Bhaskara. Pell equations are extremely important in number theory, and arise in the investigation of numbers which are figurate in more than one way, for example, simultaneously square and triangular.The equation has an obvious generalization to the Pell-like equation(3)as well as the general second-order..
A -multigrade equation is a Diophantine equation of the form(1)for , ..., , where and are -vectors. Multigrade identities remain valid if a constant is added to each element of and (Madachy 1979), so multigrades can always be put in a form where the minimum component of one of the vectors is 1.Moessner and Gloden (1944) give a bevy of multigrade equations. Small-order examples are the (2, 3)-multigrade with and :(2)(3)the (3, 4)-multigrade with and :(4)(5)(6)and the (4, 6)-multigrade with and :(7)(8)(9)(10)(Madachy 1979).A spectacular example with and is given by and (Guy 1994), which has sums(11)(12)(13)(14)(15)(16)(17)(18)(19)Rivera considers multigrade equations involving primes, consecutive primes, etc.Analogous multigrade identities to Ramanujan's fourth power identity of form(20)can also be given for third and fifth powers, the former being(21)with , 2, 3, for any positive integer , and where(22)(23)and the one for fifth..
Euler conjectured that at least th powers are required for to provide a sum that is itself an th power. The conjecture was disproved by Lander and Parkin (1967) with the counterexample(1)Ekl (1998) defined an extended Euler conjecture that there are no solutions to the Diophantine equation(2)with and not necessarily distinct, such that . Defining(3)over all known solutions to equations, this conjecture asserts that . There are no known counterexamples to this conjecture (Ekl 1998). The following table gives the smallest known values of for small .min. soln.reference44.1.30Elkies (1988)55.1.40Lander et al. (1967)66.3.30Subba Rao (1934)77.4.41Ekl (1996)88.3.50S. Chase (Meyrignac)88.4.40N. Kuosa (Nov. 9, 2006; Meyrignac)99.5.51Ekl 1997 (Meyrignac)1010.6.62N. Kuosa (2002; Meyrignac)S. Chase found a 8.3.5 () solution that displaced the 8.5.5 () solution of Letac (1942). In 2006, N. Kuosa..
The Mordell conjecture states that Diophantine equations that give rise to surfaces with two or more holes have only finite many solutions in Gaussian integers with no common factors (Mordell 1922). Fermat's equation has holes, so the Mordell conjecture implies that for each integer , the Fermat equation has at most a finite number of solutions.This conjecture was proved by Faltings (1984) and hence is now also known as Falting's theorem.
Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"Solution consists..
An integer which is expressible in only one way in the form or where is relatively prime to . If the integer is expressible in more than one way, it is called a polymorph.
A Diophantine problem (i.e., one whose solution must be given in terms of integers) which seeks a solution to the following problem. Given men and a pile of coconuts, each man in sequence takes th of the coconuts left after the previous man removed his (i.e., for the first man, , for the second, ..., for the last) and gives coconuts (specified in the problem to be the same number for each man) which do not divide equally to a monkey. When all men have so divided, they divide the remaining coconuts ways (i.e., taking an additional coconuts each), and give the coconuts which are left over to the monkey. If is the same at each division, then how many coconuts were there originally? The solution is equivalent to solving the Diophantine equations(1)(2)(3)(4)(5)which can be rewritten as(6)(7)(8)(9)(10)(11)Since there are equations in the unknowns , , ..., , , and , the solutions span a one-dimensional space (i.e., there is an infinite family of solution parameterized..
A number which can be represented both in the form and in the form . This is only possible when the Pell equation(1)is solvable. Then(2)(3)
A triple of positive integers satisfying is said to be geometric if . In particular, such a triple is geometric if its terms form a geometric sequence with common ratio whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of geometric triples and the set of equivalence classes of harmonic triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .
There are two distinct entities both known as the Lagrange number. The more common one arises in rational approximation theory (Conway and Guy 1996), while the other refers to solutions of a particular Diophantine equation (Dörrie 1965).Hurwitz's irrational number theorem gives the best rational approximation possible for an arbitrary irrational number as(1)The are called Lagrange numbers, and get steadily larger for each "bad" set of irrational numbers which is excluded, as indicated in the following table.exclude1none23Lagrange numbers are of the form(2)where is a Markov number. The Lagrange numbers form a spectrum called the Lagrange spectrum.Given a Pell equation (a quadratic Diophantineequation)(3)with a quadratic surd, define(4)for each solution with . The numbers are then known as Lagrange numbers (Dörrie 1965). The product and quotient of two Lagrange numbers are also Lagrange numbers. Furthermore,..
The only whole number solution to the Diophantineequationis , . This theorem was offered as a problem by Fermat, who suppressed his own proof.
In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed number of th powers of positive integers, where is any given positive integer and depends only on . Waring originally speculated that , , and . In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacher and Toeplitz 1957, pp. 52-61).In Lagrange's four-square theorem, Lagrange proved that , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy, and Little established , and this was subsequently reduced to by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and..
It is possible to find six points in the plane, no three on a line and no four on a circle (i.e., none of which are collinear or concyclic), such that all the mutual distances are rational. An example is illustrated by Guy (1994, p. 185).It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994, p. 188).
The rational distance problem asks to find a geometric configuration satisfying given properties such that all distances along specific edges are rational numbers. (This is equivalent to having all edge lengths be integers, since the denominators of rational numbers can be cleared by multiplication.)A cuboid whose edges and face diagonals are integers is called an Euler brick. It is not known if there exists a point in a unit square all of whose distances from the corners are rational, although J. H. Conway and M. Guy found an infinite numbers of solutions to the problem of three such distances being integers, which involves solvingwhere , , and are the three distances and is the side length of the square (Guy 1994, p. 181). There are infinitely many solutions of the corresponding problem of integer distances from the corners of an equilateral triangle (Guy 1994, p. 183).In 2001, E. Pegg found a small scalene..