Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20121221 03:40:58
Wilson's TheoremWould someone please explain me the proof of Wilson's Theorem in very simple words? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #2 20121221 16:05:45
Re: Wilson's TheoremWolfram Mathworld explains: Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional). LaTex is like painting on many strips of paper and then stacking them to see what picture they make. #3 20121221 17:28:03
Re: Wilson's TheoremHow about the congruence? 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #4 20121223 04:16:40
Re: Wilson's TheoremWiki has a proof for Wilson's Theorem (Google Wilson's Theorem Proof). To follow it one must know Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional). LaTex is like painting on many strips of paper and then stacking them to see what picture they make. #5 20121223 09:50:49
Re: Wilson's TheoremThe way to prove that is divisible by for composite is as follows. If is composite and greater than 4, then can be written as a product of an integer greater or equal to 3 and an integer greater or equal to 2, i.e. where and . Now I claim that . Proof. + rearranging gives as claimed. This means that the factors of contain a sequence of consecutive integers and a nonoverlapping sequence of integers. The first sequence contains a factor of and the second sequence contains of factor of . So the product of those integers is divisible by . But the two sequences of integers are nonoverlapping and so is divisible by the product of those integers. Hence is divisible by . Last edited by scientia (20121223 09:59:05) #6 20121226 16:53:55
Re: Wilson's TheoremOkay, Wilson's theorem. Suppose is prime. We first note that . Consider the multiplicative group of modulo , where is odd. We have and . Conversely, if and , then divides divides or or . Hence only 1 and are their own multiplicative inverses modulo . It follows that . (Each number in the list multiplies by another number in the list to ≡ 1 (mod p).) Conversely suppose and . Then for some integer . Let be a prime divisor of . If were not prime, would be smaller than and so would divide and hence would divide , a contradiction. Thus must be prime. Last edited by scientia (20121226 17:00:24) #7 20121226 19:07:07
Re: Wilson's TheoremWhat does multiplicative group mean 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #8 20121226 20:05:01
Re: Wilson's TheoremThe set , where is prime, forms a group under multiplication modulo . Last edited by scientia (20121226 20:05:20) #9 20121228 18:54:08
Re: Wilson's TheoremI still have a lot of confusion regarding the terminologies you used. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #10 20121228 21:03:12
Re: Wilson's TheoremSorry, I thought you understood group theory. Are you familiar with Fermat's little theorem? This states that if p is a prime and a is coprime with p, then a^{p−1} ≡ 1 (mod p). #11 20121228 21:08:48
Re: Wilson's TheoremYes I know this one though hadn't used it much 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #12 20121228 21:56:14
Re: Wilson's TheoremRight. A corollary of Fermat's little theorem is that if p is a prime and a is an integer coprime with p, then there is an integer b such that ab ≡ 1 (mod p). #13 20121228 22:07:25
Re: Wilson's TheoremIn simple words, for every integer a (mod p), b is the multiplicative inverse of a if ab congruent to 1 (mod p) 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #14 20121228 22:35:59
#15 20130101 17:48:10
Re: Wilson's TheoremSo the proof basically is that all other numbers are congruent to 1 mod p except (p1)*(1) is congruent to 1 mod p. And multiplying together, we get (p1)! congruent to 1 mod p. 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda #16 20130101 19:52:13
Re: Wilson's TheoremNo, all the other numbers are not congrent to 1 mod p. It is the product of them and their inverses (which are distinct from themselves) which are congruent to 1 mod p. And so on. Last edited by scientia (20130101 19:54:56) #17 20130102 00:34:13
Re: Wilson's TheoremI see 'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda 