A note written in 1637 in the margin of a book says that there are no positive integers x, y and z such that xn + yn = zn, where n is an integer greater than two. This wasn't proved until 1995. Who is the author of this note?
"as the 2016 Abel Prize. When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a ""stunning proof"". Fermat's Last Theorem, formulated in 1637, states that no three distinct positive integers ""a"", ""b"", and ""c"" can satisfy the equation if ""n"" is an integer greater than two (""n"" > 2). Over time, this simple assertion became one of the most famous unproved claims in mathematics. Between its publication and Andrew Wiles' eventual solution over 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values"
"states that for no integer formula_1 are there positive integer values of ""x"", ""y"", and ""z"" such that formula_2. Before this was proven for all formula_1 in 1995 by Andrew Wiles, it had been proven for many values of ""n"". These proofs inspired further research in the area, and formed a quasi-empirical consensus for the theorem. Even though such knowledge is more conjectural than a strictly proven theorem, it was still used in developing other mathematical ideas. Putnam has contributed to scientific fields not directly related to his work in philosophy. As a mathematician, Putnam contributed to the resolution of"
"3 × 7 × 13 = 273. Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree integers. The central binomial coefficient is never squarefree for ""n"" > 4. This was proven in 1985 for all sufficiently large integers by András Sárközy, and for all integers > 4 in 1996 by Olivier Ramaré and Andrew Granville. The multiplicative function formula_34 is defined to map positive integers ""n"" to ""t""-free numbers by reducing the exponents in the prime power representation modulo ""t"": The value set of formula_36, in particular, are the square-free"
"Fermat's Last Theorem In number theory Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known to have infinitely many solutions since antiquity. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of ""Arithmetica"" where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in"
"of ""n"" greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of ""Arithmetica"" where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics"
"a positive integer then for some rational number ""p""/""q"". Specifically, writing the infinite series on the left as ζ(2""n"") he showed where the ""B"" are the rational Bernoulli numbers. Once it was proved that π is always irrational this showed that ζ(2""n"") is irrational for all positive integers ""n"". No such representation in terms of π is known for the so-called zeta constants for odd arguments, the values ζ(2""n""+1) for positive integers ""n"". It has been conjectured that the ratios of these quantities are transcendental for every integer ""n"" ≥ 1. Because of this, no proof could be found to"
"integer which is neither of the form 8""n"" + 7, nor of the form 4""n"", is the sum of three squares, but did not provide a satisfactory proof. In 1796 Gauss proved his Eureka theorem that every positive integer ""n"" is the sum of 3 triangular numbers; this is equivalent to the fact that 8""n"" + 3 is a sum of three squares. In 1797 or 1798 A.-M. Legendre obtained the first proof of his 3 square theorem. In 1813, A. L. Cauchy noted that Legendre's theorem is equivalent to the statement in the introduction above. Previously, in 1801, C."
"15 and 290 theorems The 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000. In 2005, Bhargava and Jonathan P. Hanke announced a proof of Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290. The proof is to appear"
"of 2018, the proof is widely accepted in the mathematics community, but it has not yet been published in a peer-reviewed journal. Some state the conjecture as This version excludes 7 = 2+2+3 because this requires the even prime 2. On odd numbers larger than 7 it is slightly stronger as it also excludes sums like 17 = 2+2+13, which are allowed in the other formulation. Helfgott's proof covers both versions of the conjecture. Like the other formulation, this one also immediately follows from Goldbach's strong conjecture. In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the"
"century by a story involving Srinivasa Ramanujan. In 1938, G. H. Hardy and E. M. Wright proved that such numbers exist for all positive integers ""n"", and their proof is easily converted into a program to generate such numbers. However, the proof makes no claims at all about whether the thus-generated numbers are ""the smallest possible"" and thus it cannot be used to find the actual value of Ta(""n""). The restriction of the summands to positive numbers is necessary, because allowing negative numbers allows for more (and smaller) instances of numbers that can be expressed as sums of cubes in"
"1}, {""x"", ""x"" + 2}, and {""x"", 2""x"" + 1}. Bunyakovsky conjecture The Bunyakovsky conjecture (or Bouniakowsky conjecture) stated in 1857 by the Russian mathematician Viktor Bunyakovsky, asserts when a polynomial formula_1 in one variable with positive degree and integer coefficients should have infinitely many prime values for positive integer inputs. Three necessary conditions are Bunyakovsky's conjecture is that these three conditions are sufficient: if formula_1 satisfies the three conditions then formula_4 is prime for infinitely many positive integers formula_3. For example, all cyclotomic polynomials are irreducible, with positive (in fact, =1) leading coefficient, and as formula_9 runs over the"
"instance there are only seven distinct solutions for ""n"" = 73. The restriction that ""x"", ""y"", and ""z"" be positive is essential to the difficulty of the problem, for if negative values were allowed the problem could be solved trivially via one of the two identities and Alternatively, for any odd ""n"", a three-term solution with one negative term is possible: A generalized version of the conjecture states that, for any positive ""k"" there exists a number ""N"" such that, for all ""n"" ≥ ""N"", there exists a solution in positive integers to ""k""/""n"" = 1/""x"" + 1/""y"" + 1/""z""."
"in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but he never published it. For each of the values of ""n"" from 2 to 30, the following table shows the number (""n"" − 1)! and the remainder when (""n"" − 1)! is divided by ""n"". (In the notation of modular arithmetic, the remainder when ""m"" is divided by ""n"" is written ""m"" mod ""n"".) The background color is blue for prime values of ""n"", gold for composite values. Both of the proofs (for prime moduli) below make use of the fact that the"
"of a number is referred to as its sign. Every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers (together with zero) are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers appeared for the first time in history in the ""Nine Chapters on the Mathematical Art"", which in its present form dates from the period of the Chinese Han Dynasty (202"
"modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625. The statement that every prime ""p"" of the form ""4n+1"" is the sum of two squares is sometimes called ""Girard's theorem"". For his part, Fermat wrote an elaborate"
"1.9287800... with the property that are all prime. Let π(""x"") be the number of primes less than or equal to ""x"". It is a straightforward deduction from Wilson's theorem that Also, if ""n"" ≥ 2, None of the formulas in this section are of any practical use. Ramanujan submitted these problems to the ""Journal of the Indian Mathematical Society"". If ""n"" is a positive integer, prove that (i) formula_101 (ii) formula_102 (iii) formula_103 The study of Waring's problem has led to an unsolved problem: Are there any positive integers ""k"" ≥ 6 such that Mahler has proved there can only"
"primes begins 2, 3, 2, 5, 2, 3, 7, 2 . Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers ""n"" as maxima of the above function. In the 1980s Guy Robin showed that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all ""n"" > 5040: (where γ is the Euler–Mascheroni constant) This inequality is known to fail for 27 numbers : Robin showed that if the Riemann hypothesis is true then ""n"" = 5040 is the last integer for which it fails. The inequality is"
"positive integer is either a cube or the sum of not more than nine cubes. He also advanced the hypothesis that every positive integer is either a biquadrate (fourth power) or the sum of not more than nineteen biquadrates. These hypotheses form what is known as Waring's problem. He also published a theorem, due to his friend John Wilson, concerning prime numbers; it was later proven rigorously by Lagrange. In ""Proprietates Algebraicarum Curvarum"" (1772) Waring reissued in a much revised form the first four chapters of the second part of ""Miscellanea Analytica"". He devoted himself to the classification of higher"
"""n"" variables, ""n"" ≥ 5, nontrivially represents zero, i.e. there exists a non-zero vector ""x"" with integer components such that ""Q""(""x"") = 0. The Oppenheim conjecture can be viewed as an analogue of this statement for forms ""Q"" that are not multiples of a rational form. It states that in this case, the set of values of ""Q"" on integer vectors is a dense subset of the real line. Several versions of the conjecture were formulated by Oppenheim and Davenport. For ""n"" ≥ 5 this was conjectured by Oppenheim in 1929; the stronger version is due to Davenport in 1946."
"years. One may easily show that there is not any other solution with and positive integers less than 10. Many well known puzzles in the field of recreational mathematics lead to diophantine equations. Examples include the Cannonball problem, Archimedes's cattle problem and The monkey and the coconuts. In 1637, Pierre de Fermat scribbled on the margin of his copy of ""Arithmetica"": ""It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers."" Stated in more modern language, ""The equation has"
"Bunyakovsky conjecture The Bunyakovsky conjecture (or Bouniakowsky conjecture) stated in 1857 by the Russian mathematician Viktor Bunyakovsky, asserts when a polynomial formula_1 in one variable with positive degree and integer coefficients should have infinitely many prime values for positive integer inputs. Three necessary conditions are Bunyakovsky's conjecture is that these three conditions are sufficient: if formula_1 satisfies the three conditions then formula_4 is prime for infinitely many positive integers formula_3. For example, all cyclotomic polynomials are irreducible, with positive (in fact, =1) leading coefficient, and as formula_9 runs over the positive integers, formula_10 don't share a common factor greater than"
"positive integer . The above proof is a polished version, which is kept as simple as possible concerning the prerequisites, of an analysis of the formula which is obtained by integrations by parts. Claim 2 essentially establishes this formula, where the use of ""F"" hides the iterated integration by parts. The last integral vanishes because is the zero polynomial. Claim 1 shows that the remaining sum is an integer. Niven's proof is closer to Cartwright's (and therefore Hermite's) proof than it appears at first sight. In fact, Therefore, the substitution ""xz"" = ""y"" turns this integral into In particular, Another"
"Equation xʸ=yˣ In general, exponentiation fails to be commutative. However, the equation formula_1 holds in special cases, such as formula_2 The equation formula_1 is mentioned in a letter of Bernoulli to Goldbach (29 June 1728). The letter contains a statement that when formula_4 the only solutions in natural numbers are formula_5 and formula_6 although there are infinitely many solutions in rational numbers. The reply by Goldbach (31 January 1729) contains general solution of the equation obtained by substituting formula_7 A similar solution was found by Euler. J. van Hengel pointed out that if formula_8 are positive integers with formula_9 then"
"Apéry's theorem In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number cannot be written as a fraction ""p""/""q"" with ""p"" and ""q"" being integers. The special values of the Riemann zeta function at even integers ""2n"" (""n > 0"") can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function's values are in general rational or not at the odd integers ""2n + 1"" (""n"" > 1) (though they are conjectured to be irrational). Euler proved that if ""n"" is"
"≤ 1685 or higher, there appears a second solution ""X"" = 4, ""Y"" = 61. Thus, from then on, the problem is not solvable in the sense that there is no longer a unique solution. Similarly, if ""X"" + ""Y"" ≤ 1970 or higher a third solution appears (""X"" = 16, ""Y"" = 73). All of these three solutions contain one prime number. The first solution with no prime number is the fourth which appears at ""X"" + ""Y"" ≤ 2522 or higher with values ""X"" = 16 = 2·2·2·2 and ""Y"" = 111 = 3·37. If the condition ""Y"""
"arguments. Ramanujan's ternary quadratic form In mathematics, in number theory, Ramanujan's ternary quadratic form is the algebraic expression with integral values for ""x"", ""y"" and ""z"". Srinivasa Ramanujan considered this expression in a footnote in a paper published in 1916 and briefly discussed the representability of integers in this form. After giving necessary and sufficient conditions that an integer cannot be represented in the form for certain specific values of ""a"", ""b"" and ""c"", Ramanujan observed in a footnote: ""(These) results may tempt us to suppose that there are similar simple results for the form whatever are the values of"
"no primitive solutions in integers (no pairwise coprime solutions). In turn, this proves Fermat's Last Theorem for the case ""n"" = 4, since the equation ""a"" + ""b"" = ""c"" can be written as ""a"" + ""b"" = (""c""). Alternative proofs of the case ""n"" = 4 were developed later by Frénicle de Bessy (1676), Leonhard Euler (1738), Kausler (1802), Peter Barlow (1811), Adrien-Marie Legendre (1830), Schopis (1825), Terquem (1846), Joseph Bertrand (1851), Victor Lebesgue (1853, 1859, 1862), Theophile Pepin (1883), Tafelmacher (1893), David Hilbert (1897), Bendz (1901), Gambioli (1901), Leopold Kronecker (1901), Bang (1905), Sommer (1907), Bottari (1908), Karel"
"only if either (a) ""n"" = 2 or (b) 2""d"" = 2 or (c) ""n"" = 3 and 2""d"" = 4. Hilbert's proof did not exhibit an explicit example: only in 1967 the first explicit example was constructed by Motzkin. The particular case of ""n"" = 2 was already solved by Hilbert in 1893. The general problem was solved in the affirmative, in 1927, by Emil Artin, for positive semidefinite functions over the reals or more generally real-closed fields. An algorithmic solution was found by Charles Delzell in 1984. A result of Albrecht Pfister shows that a positive semidefinite form"
"""n"" = 7. Proof of Fermat's Last Theorem for specific exponents Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent ""n"" larger than 2. In the centuries following the initial statement of the result and its general proof, various proofs were devised for particular values of the exponent ""n"". Several of these proofs are described below, including Fermat's proof in the case ""n"" = 4, which is an early example of the method of infinite descent."
"Euler, the son of Leonhard Euler, in about 1772. Later work by Dickson, Pillai, Rubugunday, Niven and many others has proved that No value of formula_1 is known for which formula_29. Mahler proved there can only be a finite number of such formula_1, and Kubina and Wunderlich have shown that any such formula_1 must satisfy formula_32 471,600,000. Thus it is conjectured that this never happens, that is, formula_33 for every positive integer formula_1. The first few values of formula_2 are: From the work of Hardy and Littlewood, the related quantity ""G""(""k"") was studied with ""g""(""k""). ""G""(""k"") is defined to be"
"to infinitely many divisibility events, one is led to guess that the probability that two numbers are coprime is given by a product over all primes, Here ""ζ"" refers to the Riemann zeta function, the identity relating the product over primes to ""ζ""(2) is an example of an Euler product, and the evaluation of ""ζ""(2) as ""π""/6 is the Basel problem, solved by Leonhard Euler in 1735. There is no way to choose a positive integer at random so that each positive integer occurs with equal probability, but statements about ""randomly chosen integers"" such as the ones above can be"
"Ramanujan's ternary quadratic form In mathematics, in number theory, Ramanujan's ternary quadratic form is the algebraic expression with integral values for ""x"", ""y"" and ""z"". Srinivasa Ramanujan considered this expression in a footnote in a paper published in 1916 and briefly discussed the representability of integers in this form. After giving necessary and sufficient conditions that an integer cannot be represented in the form for certain specific values of ""a"", ""b"" and ""c"", Ramanujan observed in a footnote: ""(These) results may tempt us to suppose that there are similar simple results for the form whatever are the values of ""a"","
"Mersenne conjectures In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one. The original, called Mersenne's conjecture, was a statement by Marin Mersenne in his ""Cogitata Physico-Mathematica"" (1644; see e.g. Dickson 1919) that the numbers formula_1 were prime for ""n"" = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers ""n"" ≤ 257. Due to the size of these numbers, Mersenne did not and could not test all of them, nor"