Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY ALGEBRA◮The prime integers p i are labelled in ascending order, thus p 1 =1, p 2 =2, p 5 =7, etc.Show that there is no largest prime number.Assume, on the contrary, that there is a largest prime and let it be p N .Considernowthenumber q formed by multiplying together all the primes from p 1 to p N and then addingone to the product, i.e.q = p 1 p 2 ···p N +1.By our assumption p N is the largest prime, and so no number can have a prime factorgreater than this. However, for every prime p i , i =1, 2,...,N, the quotient q/p i has theform M i +(1/p i )withM i an integer and 1/p i non-integer. This means that q/p i cannot bean integer and so p i cannot be a divisor of q.Since q is not divisible by any of the (assumed) finite set of primes, it must be itself aprime. As q is also clearly greater than p N , we have a contradiction. This shows that ourassumption that there is a largest prime integer must be false, and so it follows that thereis no largest prime integer.It should be noted that the given construction for q does not generate all the primesthat actually exist (e.g. for N =3,q = 7 rather than the next actual prime value of 5, isfound), but this does not matter for the purposes of our proof by contradiction. ◭1.7.3 Necessary and sufficient conditionsAs the final topic in this introductory chapter, we consider briefly the notionof, and distinction between, necessary and sufficient conditions in the contextof proving a mathematical proposition. In ordinary English the distinction iswell defined, and that distinction is maintained in mathematics. However, inthe authors’ experience students tend to overlook it and assume (wrongly) that,having proved that the validity of proposition A implies the truth of propositionB, it follows by ‘reversing the argument’ that the validity of B automaticallyimplies that of A.As an example, let proposition A be that an integer N is divisible withoutremainder by 6, and proposition B be that N is divisible without remainder by2. Clearly, if A is true then it follows that B is true, i.e. A is a sufficient conditionfor B; it is not however a necessary condition, as is trivially shown by taking Nas 8. Conversely, the same value of N shows that whilst the validity of B is anecessary condition for A to hold, it is not sufficient.An alternative terminology to ‘necessary’ and ‘sufficient’ often employed bymathematicians is that of ‘if’ and ‘only if’, particularly in the combination ‘if andonly if’ which is usually written as IFF or denoted by a double-headed arrow⇐⇒ . The equivalent statements can be summarised byA if B A is true if B is true or B =⇒ A,B is a sufficient condition for A B =⇒ A,A only if B A is true only if B is true or A =⇒ B,B is a necessary consequence of A A =⇒ B,34