The number of different triangles which have integer side lengths and perimeter is(1)(2)(3)where is the partition function giving the number of ways of writing as a sum of exactly terms, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly complicated closed form is given by(4)The values of for , 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.The generating function for is given by(5)(6)(7) also satisfies(8)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 shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational..
A twin Pythagorean triple is a Pythagorean triple for which two values are consecutive integers. By definition, twin triplets are therefore primitive triples. Of the 16 primitive triples with hypotenuse less than 100, seven are twin triples. The first few twin triples, sorted by increasing , are (3, 4, 5), (5, 12, 13), (7, 24, 25), (20, 21, 29), (9, 40, 41), (11, 60, 61), (13, 84, 85), (15, 112, 113), ....The numbers of twin triples with hypotenuse less than 10, , , ... are 1, 7, 24, 74, ... (OEIS A101903).The first few leg-leg triplets are (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985), ... (OEIS A001652, A046090, and A001653). A closed form is available for the th such pair. Consider the general reduced solution , then the requirement that the legs be consecutive integers is(1)Rearranging gives(2)Defining(3)(4)then gives the Pell equation(5)Solutions to the Pell equation are given by(6)(7)so the lengths of the legs and and the hypotenuse..
Lucas's theorem states that if be a squarefree integer and a cyclotomic polynomial, then(1)where and are integer polynomials of degree and , respectively. This identity can be expressed as(2)with and symmetric polynomials. The following table gives the first few and s (Riesel 1994, pp. 443-456).213156710
An idoneal number, also called a suitable number or convenient number, is a positive integer for which the fact that a number is a monomorph (i.e., is expressible in only one way as where is relatively prime to ) guarantees it to be a prime, prime power, or twice one of these. The numbers are also called Euler's idoneal numbers or suitable numbers.A positive integer is idoneal iff it cannot be written as for integer , , and with .The 65 idoneal numbers found by Gauss and Euler and conjectured to be the only such numbers (Shanks 1969) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, and 1848 (OEIS A000926). It is known that if any other idoneal numbers exist, there can be only one more...
A Heronian tetrahedron, also called a perfect tetrahedron, is a (not necessarily regular) tetrahedron whose sides, face areas, and volume are all rational numbers. It therefore is a tetrahedron all of whose faces are Heronian triangles and additionally that has rational volume. (Note that the volume of a tetrahedron can be computed using the Cayley-Menger determinant.)The integer Heronian tetrahedron having smallest maximum side length is the one with edge lengths 51, 52, 53, 80, 84, 117; faces (117, 80, 53), (117, 84, 51), (80, 84, 52), (53, 51, 52); face areas 1170, 1800, 1890, 2016; and volume 18144 (Buchholz 1992; Guy 1994, p. 191). This is the only integer Heronian triangle with all side lengths less than 157.The integer Heronian tetrahedron with smallest possible surface area and volume has edges 25, 39, 56, 120, 153, and 160; areas 420, 1404, 1872, and 2688 (for a total surface area of 6384); and volume 8064 (Buchholz 1992, Peterson..
A Pythagorean triple is a triple of positive integers , , and such that a right triangle exists with legs and hypotenuse . By the Pythagorean theorem, this is equivalent to finding positive integers , , and satisfying(1)The smallest and best-known Pythagorean triple is . The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.Plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of and , and are therefore symmetric about both the x- and y-axes.Similarly, plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds.It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which and are relatively prime, since other solutions can be generated trivially from the primitive ones. The primitive triples are illustrated above, and..
A Pythagorean triangle is a right triangle with integer side lengths (i.e., whose side lengths form a Pythagorean triple). A Pythagorean triangle with is known as a primitive right triangle.The inradius of a Pythagorean triangle is always a whole number sinceThe area of such a triangle is also a whole number since for primitive Pythagorean triples, one of or must be even, and for imprimitive triples, both and are even, sois always a positive integer.
is prime iff the 14 Diophantine equations in 26 variables(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)have a solution in positive integers (Jones etal. 1976; Riesel 1994, p. 40).
Let there be integers with . The values represent the denominations of different coins, where these denominations have greatest common divisor of 1. The sums of money that can be represented using the given coins are then given by(1)where the are nonnegative integers giving the numbers of each coin used. If , it is obviously possibly to represent any quantity of money . However, in the general case, only some quantities can be produced. For example, if the allowed coins are , it is impossible to represent and 3, although all other quantities can be represented.Determining the function giving the greatest for which there is no solution is called the coin problem, or sometimes the money-changing problem. The largest such for a given problem is called the Frobenius number .The result(2)(3)(Nijenhuis and Wilf 1972) is mathematical folklore. The total number of such nonrepresentable amounts is given by(4)The largest nonrepresentable amounts for..
Fermat's sandwich theorem states that 26 is the only number sandwiched between a perfect square number ( and a perfect cubic number (). According to Singh (1997), after challenging other mathematicians to establish this result while not revealing his own proof, Fermat took particular delight in taunting the English mathematicians Wallis and Digby with their inability to prove the result.
A perfect cuboid is a cuboid having integer side lengths,integer face diagonals(1)(2)(3)and an integer space diagonal(4)The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.No perfect cuboids are known despite an exhaustive search for all "odd sides" up to (Butler, pers. comm., Dec. 23, 2004).Solving the perfect cuboid problem is equivalent to solving the Diophantineequations(5)(6)(7)(8)A solution with integer space diagonal and two out of three face diagonals is , , and , giving , , , and , which was known to Euler. A solution giving integer space and face diagonals with only a single nonintegral polyhedron edge is , , and , giving , , , and .
An Euler brick is a cuboid that possesses integer edges and face diagonals(1)(2)(3)If the space diagonal is also an integer, the Euler brick is called a perfect cuboid, although no examples of perfect cuboids are currently known.The smallest Euler brick has sides and face polyhedron diagonals , , and , and was discovered by Halcke (1719; Dickson 2005, pp. 497-500). Kraitchik gave 257 cuboids with the odd edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003 smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (OEIS A031173, A031174, and A031175).Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution always giving Euler bricks (but not giving all possible Euler bricks), while in 1770 and 1772, Euler found at least two parametric solutions...
The Frobenius number is the largest value for which the Frobenius equation(1)has no solution, where the are positive integers, is an integer, and the solutions are nonnegative integer. As an example, if the values are 4 and 9, then 23 is the largest unsolvable number. Similarly, the largest number that is not a McNugget number (a number obtainable by adding multiples of 6, 9, and 20) is 43.Finding the Frobenius number of a given problem is known as the coinproblem.Computation of the Frobenius number is implemented in the Wolfram Language as FrobeniusNumber[a1, ..., an].Sylvester (1884) showed(2)(3)
Given a Pythagorean triple , the fractions and are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of fractions of the form .
Let be a prime number, thenwhere and are homogeneous polynomials in and with integer coefficients. Gauss (1965, p. 467) gives the coefficients of and up to .Kraitchik (1924) generalized Gauss's formula to odd squarefree integers . Then Gauss's formula can be written in the slightly simpler formwhere and have integer coefficients and are of degree and , respectively, with the totient function and a cyclotomic polynomial. In addition, is symmetric if is even; otherwise it is antisymmetric. is symmetric in most cases, but it antisymmetric if is of the form (Riesel 1994, p. 436). The following table gives the first few and s (Riesel 1994, pp. 436-442).51711
The Frobenius equation is the Diophantine equationwhere the are positive integers, is an integer, and the solutions are nonnegative integers. Solution of the Frobenius equation is implemented using FrobeniusSolve[a1, ..., an, b].The largest value for which the Frobenius equation has no solution is known as the Frobenius number.
Consider a set of positive integer-denomination postage stamps sorted such that . Suppose they are to be used on an envelope with room for no more than stamps. The postage stamp problem then consists of determining the smallest integer which cannot be represented by a linear combination with and .Without the latter restriction, this problem is known as the Frobenius problem or Frobenius postage stamp problem.The number of consecutive possible postage amounts is given by(1)where is called an -range.Exact solutions exist for arbitrary for and 3. The solution is(2)for . It is also known that(3)(Stöhr 1955, Guy 1994), where is the floor function, the first few values of which are 2, 4, 7, 10, 14, 18, 23, 28, 34, 40, ... (OEIS A014616; Guy 1994, p. 123).Hofmeister (1968, 1983) showed that for ,(4)where and are functions of (mod 9), and Mossige (1981, 1987) showed that(5)(Guy 1994, p. 123).Shallit (2002) proved that the (local) postage..