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46-417: The chocolate-bar formulation of Chomp is due to David Gale , but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh . Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element (poisonous block) removed. Below shows the sequence of moves in

92-429: A and b , where n = a b , and 1 < a ≤ b < n . By the induction hypothesis, a = p 1 p 2 ⋅⋅⋅ p j and b = q 1 q 2 ⋅⋅⋅ q k are products of primes. But then n = a b = p 1 p 2 ⋅⋅⋅ p j q 1 q 2 ⋅⋅⋅ q k is a product of primes. Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let n be

138-468: A and b themselves: However, integer factorization , especially of large numbers, is much more difficult than computing products, GCDs, or LCMs. So these formulas have limited use in practice. Many arithmetic functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers. The proof uses Euclid's lemma ( Elements VII, 30): If

184-560: A composite number greater than 1 {\displaystyle 1} . Now, say Every p i {\displaystyle p_{i}} must be distinct from every q j . {\displaystyle q_{j}.} Otherwise, if say p i = q j , {\displaystyle p_{i}=q_{j},} then there would exist some positive integer t = s / p i = s / q j {\displaystyle t=s/p_{i}=s/q_{j}} that

230-645: A field . However, the theorem does not hold for algebraic integers . This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem . The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between Fermat's statement and Wiles's proof . The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid 's Elements . If two numbers by multiplying one another make some number, and any prime number measure

276-406: A least common multiple of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by André Weil . Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. While Euclid took the first step on

322-455: A 5 × 4 bar, at least one of A's moves is a mistake. The intermediate positions in an m × n Chomp are integer-partitions (non-increasing sequences of positive integers) λ 1 ≥ λ 2 ≥···≥ λ r , with λ 1 ≤ n and r ≤ m . Their number is the binomial coefficient ( m + n n ) {\displaystyle {\binom {m+n}{n}}} , which grows exponentially with m and n . Chomp belongs to

368-400: A prime divides the product of two integers, then it must divide at least one of these integers. It must be shown that every integer greater than 1 is either prime or a product of primes. First, 2 is prime. Then, by strong induction , assume this is true for all numbers greater than 1 and less than n . If n is prime, there is nothing more to prove. Otherwise, there are integers

414-572: A prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers Z {\displaystyle \mathbb {Z} } every irreducible is prime". This is also true in Z [ i ] {\displaystyle \mathbb {Z} [i]} and Z [ ω ] , {\displaystyle \mathbb {Z} [\omega ],} but not in Z [ − 5 ] . {\displaystyle \mathbb {Z} [{\sqrt {-5}}].} The rings in which factorization into irreducibles

460-512: A product of primes ( up to the order and multiplication by units). Similarly, in 1844 while working on cubic reciprocity , Eisenstein introduced the ring Z [ ω ] {\displaystyle \mathbb {Z} [\omega ]} , where ω = − 1 + − 3 2 , {\textstyle \omega ={\frac {-1+{\sqrt {-3}}}{2}},}   ω 3 = 1 {\displaystyle \omega ^{3}=1}

506-430: A product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, 12 = 2 ⋅ 6 = 3 ⋅ 4 {\displaystyle 12=2\cdot 6=3\cdot 4} ). This theorem

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552-536: A product, for example, 2 = ab , then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + √ −5 ) nor (1 − √ −5 ) even though it divides their product 6. In algebraic number theory 2 is called irreducible in Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} (only divisible by itself or

598-516: A smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer 1 {\displaystyle 1} , not factor into any prime. The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity . This paper introduced what

644-502: A special case of the unique factorization theorem in commutative Möbius monoids. The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections:

690-438: A standard reference for this area. The Gale transform is an involution on sets of points in projective space . The concept is important in optimization , coding theory , and algebraic geometry . Gale's 1962 paper with Lloyd Shapley on the stable marriage problem provides the first formal statement and proof of a problem that has far-reaching implications in many matching markets. The resulting Gale–Shapley algorithm

736-431: A typical game starting with a 5 × 4 bar: Player A eats two blocks from the bottom right corner; Player B eats three from the bottom row; Player A picks the block to the right of the poisoned block and eats eleven blocks; Player B eats three blocks from the remaining column, leaving only the poisoned block. Player A must eat the last block and so loses. Note that since it is provable that player A can win when starting from

782-541: A unit) but not prime in Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} (if it divides a product it must divide one of the factors). The mention of Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} is required because 2 is prime and irreducible in Z . {\displaystyle \mathbb {Z} .} Using these definitions it can be proven that in any integral domain

828-401: Is a cube root of unity . This is the ring of Eisenstein integers , and he proved it has the six units ± 1 , ± ω , ± ω 2 {\displaystyle \pm 1,\pm \omega ,\pm \omega ^{2}} and that it has unique factorization. However, it was also discovered that unique factorization does not always hold. An example

874-424: Is a notable open problem; a $ 100 reward has been offered for finding a winning first move. More generally, Chomp can be played on any partially ordered set with a least element . A move is to remove any element along with all larger elements. A player loses by taking the least element. All varieties of Chomp can also be played without resorting to poison by using the misère play convention : The player who eats

920-515: Is also impossible, as, if p 1 {\displaystyle p_{1}} is a divisor of q 1 − p 1 , {\displaystyle q_{1}-p_{1},} it must be also a divisor of q 1 , {\displaystyle q_{1},} which is impossible as p 1 {\displaystyle p_{1}} and q 1 {\displaystyle q_{1}} are distinct primes. Therefore, there cannot exist

966-677: Is currently being applied in New York and Boston public school systems in assigning students to schools. In 2012 The Nobel Prize in Economics was awarded to Shapley for this work. Gale wrote a Mathematical Entertainments column for The Mathematical Intelligencer from 1991 through 1997. The book Tracking the Automatic Ant collects these columns. In 2004 Gale developed MathSite, a pedagogic website that uses interactive exhibits to illustrate important mathematical ideas. MathSite won

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1012-435: Is divided evenly by some prime number.) Proposition 31 is proved directly by infinite descent . Any number either is prime or is measured by some prime number. Proposition 32 is derived from proposition 31, and proves that the decomposition is possible. If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it. (In modern terminology:

1058-408: Is essentially unique are called unique factorization domains . Important examples are polynomial rings over the integers or over a field , Euclidean domains and principal ideal domains . In 1843 Kummer introduced the concept of ideal number , which was developed further by Dedekind (1876) into the modern theory of ideals , special subsets of rings. Multiplication is defined for ideals, and

1104-412: Is given by Z [ − 5 ] {\displaystyle \mathbb {Z} [{\sqrt {-5}}]} . In this ring one has Examples like this caused the notion of "prime" to be modified. In Z [ − 5 ] {\displaystyle \mathbb {Z} \left[{\sqrt {-5}}\right]} it can be proven that if any of the factors above can be represented as

1150-442: Is now called the ring of Gaussian integers , the set of all complex numbers a + bi where a and b are integers. It is now denoted by Z [ i ] . {\displaystyle \mathbb {Z} [i].} He showed that this ring has the four units ±1 and ± i , that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as

1196-514: Is one of the main reasons why 1 is not considered a prime number : if 1 were prime, then factorization into primes would not be unique; for example, 2 = 2 ⋅ 1 = 2 ⋅ 1 ⋅ 1 = … {\displaystyle 2=2\cdot 1=2\cdot 1\cdot 1=\ldots } The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains , Euclidean domains , and polynomial rings over

1242-847: Is smaller than s and has two distinct prime factorizations. One may also suppose that p 1 < q 1 , {\displaystyle p_{1}<q_{1},} by exchanging the two factorizations, if needed. Setting P = p 2 ⋯ p m {\displaystyle P=p_{2}\cdots p_{m}} and Q = q 2 ⋯ q n , {\displaystyle Q=q_{2}\cdots q_{n},} one has s = p 1 P = q 1 Q . {\displaystyle s=p_{1}P=q_{1}Q.} Also, since p 1 < q 1 , {\displaystyle p_{1}<q_{1},} one has Q < P . {\displaystyle Q<P.} It then follows that As

1288-549: The n i are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0 ). This representation is called the canonical representation of n , or the standard form of n . For example, Factors p = 1 may be inserted without changing the value of n (for example, 1000 = 2 ×3 ×5 ). In fact, any positive integer can be uniquely represented as an infinite product taken over all

1334-441: The L , the first player replies with the same move on the second arm, always presenting the second player again with a symmetric L shape. Finally, this L will degenerate into the single poisonous square, and the second player would lose. Three- dimensional Chomp has an initial chocolate bar of a cuboid of blocks indexed as (i,j,k). A move is to take a block together with any block all of whose indices are greater or equal to

1380-451: The dimensions of the Chomp board are given by the exponents of the primes in its prime factorization . Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers : for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block. The case of ω × ω × ω Chomp

1426-573: The 2007 Pirelli Internetional Award for Science Communication in Mathematics. Fundamental theorem of arithmetic In mathematics , the fundamental theorem of arithmetic , also called the unique factorization theorem and prime factorization theorem , states that every integer greater than 1 can be represented uniquely as a product of prime numbers , up to the order of the factors. For example, The theorem says two things about this example: first, that 1200 can be represented as

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1472-416: The bottom right hand square. By our assumption, the second player has a response to this which will force victory. But if such a winning response exists, the first player could have played it as their first move and thus forced victory. The second player therefore cannot have a winning strategy. Computers can easily calculate winning moves for this game on two-dimensional boards of reasonable size. However, as

1518-477: The category of impartial two-player perfect information games, making it also analyzable by Nim because of the Sprague–Grundy theorem . For any rectangular starting position, other than 1×1, the first player can win. This can be shown using a strategy-stealing argument : assume that the second player has a winning strategy against any initial first-player move. Suppose then, that the first player takes only

1564-444: The corresponding index of the chosen block. In the same way Chomp can be generalised to any number of dimensions. Chomp is sometimes described numerically. An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models n- dimensional Chomp, where the initial natural number has n prime factors and

1610-523: The faculty at the University of California, Berkeley . Gale lived in Berkeley, California , and Paris , France with his partner Sandra Gilbert , feminist literary scholar and poet. He has three daughters and two grandsons. Gale's contributions to mathematical economics include an early proof of the existence of competitive equilibrium , his solution of the n -dimensional Ramsey problem , in

1656-408: The final chocolate block is not poisoned, but simply loses by virtue of being the last player. This is identical to the ordinary rule when playing Chomp on its own, but differs when playing the disjunctive sum of Chomp games, where only the last final chocolate block loses. David Gale David Gale (December 13, 1921 – March 7, 2008) was an American mathematician and economist . He

1702-691: The least such integer and write n = p 1 p 2 ... p j = q 1 q 2 ... q k , where each p i and q i is prime. We see that p 1 divides q 1 q 2 ... q k , so p 1 divides some q i by Euclid's lemma . Without loss of generality, say p 1 divides q 1 . Since p 1 and q 1 are both prime, it follows that p 1 = q 1 . Returning to our factorizations of n , we may cancel these two factors to conclude that p 2 ... p j = q 2 ... q k . We now have two distinct prime factorizations of some integer strictly smaller than n , which contradicts

1748-520: The minimality of n . The fundamental theorem of arithmetic can also be proved without using Euclid's lemma. The proof that follows is inspired by Euclid's original version of the Euclidean algorithm . Assume that s {\displaystyle s} is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that s {\displaystyle s} , if it exists, must be

1794-405: The number of positions grows exponentially, this is infeasible for larger boards. For a square starting position (i.e., n × n for any n ≥ 2), the winning strategy can easily be given explicitly. The first player should present the second with an L shape of one row and one column only, of the same length, connected at the poisonous square. Then, whatever the second player does on one arm of

1840-579: The positive integers less than s have been supposed to have a unique prime factorization, p 1 {\displaystyle p_{1}} must occur in the factorization of either q 1 − p 1 {\displaystyle q_{1}-p_{1}} or Q . The latter case is impossible, as Q , being smaller than s , must have a unique prime factorization, and p 1 {\displaystyle p_{1}} differs from every q j . {\displaystyle q_{j}.} The former case

1886-423: The positive prime numbers, as where a finite number of the n i are positive integers, and the others are zero. Allowing negative exponents provides a canonical form for positive rational numbers . The canonical representations of the product, greatest common divisor (GCD), and least common multiple (LCM) of two numbers a and b can be expressed simply in terms of the canonical representations of

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1932-413: The product, it will also measure one of the original numbers. (In modern terminology: if a prime p divides the product ab , then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma , and it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. (In modern terminology: every integer greater than one

1978-443: The rings in which they have unique factorization are called Dedekind domains . There is a version of unique factorization for ordinals , though it requires some additional conditions to ensure uniqueness. Any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Fundamental Theorem of Arithmetic is, in fact,

2024-482: The theory of optimal economic growth. Gale and F. M. Stewart initiated the study of infinite games with perfect information . This work led to fundamental contributions to mathematical logic . Gale is the inventor of the game of Bridg-It (also known as "Game of Gale") and Chomp . Gale played a fundamental role in the development of the theory of linear programming and linear inequalities. His classic 1960 book The Theory of Linear Economic Models continues to be

2070-470: The way to the existence of prime factorization, Kamāl al-Dīn al-Fārisī took the final step and stated for the first time the fundamental theorem of arithmetic. Article 16 of Gauss 's Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic . Every positive integer n > 1 can be represented in exactly one way as a product of prime powers where p 1 < p 2 < ... < p k are primes and

2116-530: Was a professor emeritus at the University of California, Berkeley , affiliated with the departments of mathematics, economics, and industrial engineering and operations research. He has contributed to the fields of mathematical economics , game theory , and convex analysis . Gale earned his B.A. from Swarthmore College , obtained an M.A. from the University of Michigan in 1947, and earned his Ph.D. in Mathematics at Princeton University in 1949. He taught at Brown University from 1950 to 1965 and then joined

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