Bell numbers, partition moves and the eigenvalues of the random
... We represent partitions by Young diagrams. Motivated by the Branching Rule for irreducible representations of symmetric groups, we say that a box in a Young diagram is removable if removing it leaves the Young diagram of a partition; a position to which a box may be added to give a Young diagram of ...
... We represent partitions by Young diagrams. Motivated by the Branching Rule for irreducible representations of symmetric groups, we say that a box in a Young diagram is removable if removing it leaves the Young diagram of a partition; a position to which a box may be added to give a Young diagram of ...
![Maximum subsets of (0,1] with no solutions to x](http://s1.studyres.com/store/data/004884619_1-aef9649f88c5cc5f6ca0274124904717-300x300.png)
Proof by Induction
... from the contradiction that our assumption that the statement we want to prove is false is incorrect, so the statement we want to prove must be true. But that’s just silly. Why do we need the first and third steps? After all, the second step is a proof all by itself! Unfortunately, this redundant st ...
... from the contradiction that our assumption that the statement we want to prove is false is incorrect, so the statement we want to prove must be true. But that’s just silly. Why do we need the first and third steps? After all, the second step is a proof all by itself! Unfortunately, this redundant st ...
the origins of the genus concept in quadratic forms
... examples and formulated many similar conjectures (presented as theorems). It was in this paper that he also established many basic facts about quadratic residues. His most general result along these lines was the following: THEOREM 2: Let n be a nonzero integer, and let p be an odd prime relatively ...
... examples and formulated many similar conjectures (presented as theorems). It was in this paper that he also established many basic facts about quadratic residues. His most general result along these lines was the following: THEOREM 2: Let n be a nonzero integer, and let p be an odd prime relatively ...
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 a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. 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 theretofore 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 and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".