a lemma is a mathematical term. it's kind of like a mini-theorem. usually, lemmas are small bits that have proofs, and their main uses are just to prove other theorems.
an example of a lemma is the following...
Lemma: If n is a positive, composite integer, then there exists a prime number p such that p divides n.
This lemma is then used to prove the following...
Theorem: There are infinitely many primes.
Proof: (by contradiction). Suppose there are finitely many primes, p1, p2, p3,...,pn. Then let B = p1p2p3...pn + 1.
Since B > pk for all 1 <= k <= n, B cannot be prime. Hence B is composite.
So by the above lemma, there is a prime pj where 1 <= j <= n, and pj divides B.
Since pj also divides p1p2p3...pn, pj must also divide 1. (If x = y + z, where x, y, and z are integers, and a divides two of the three integers there, then a also divides the third integer too. This is also a thereom).
The only number that divides 1 is one 1 itself. Therefore pj = 1, and B must be prime too. So for any finite collection of primes, you can find another prime. Therefore, there's infinitely many primes.
an example of a lemma is the following...
Lemma: If n is a positive, composite integer, then there exists a prime number p such that p divides n.
This lemma is then used to prove the following...
Theorem: There are infinitely many primes.
Proof: (by contradiction). Suppose there are finitely many primes, p1, p2, p3,...,pn. Then let B = p1p2p3...pn + 1.
Since B > pk for all 1 <= k <= n, B cannot be prime. Hence B is composite.
So by the above lemma, there is a prime pj where 1 <= j <= n, and pj divides B.
Since pj also divides p1p2p3...pn, pj must also divide 1. (If x = y + z, where x, y, and z are integers, and a divides two of the three integers there, then a also divides the third integer too. This is also a thereom).
The only number that divides 1 is one 1 itself. Therefore pj = 1, and B must be prime too. So for any finite collection of primes, you can find another prime. Therefore, there's infinitely many primes.
Reply
Reply
Leave a comment