 Hello friends, here is another question on the application of fundamental theorem of arithmetic, which is a part of the real numbers or number theory chapter. So the question says prove that there are infinitely many positive prime numbers. So this is a proving question where you have to prove that there are infinitely many that means there are infinite numbers of positive prime numbers. So we all know what a prime number is. So a prime number is a number n which has only two divisors or factors. So n there are only two two factors are there for this natural number n one is one itself and the number itself n right. So two factors one and n if only there are two factors of any natural number then we say that that natural number n is a prime factor I'm sorry a prime number. Now the question says prove that there are infinitely many positive prime numbers that means there is no highest prime number. So you will always find a prime number greater than the previously found prime number okay. So how do we prove it? So we will be using underlying concept will be fundamental fundamental theorem of theorem of arithmetic theorem of arithmetic which says that any composite number can be expressed uniquely in terms of its prime factors as a product of its prime factors. So any any composite number a can be uniquely expressed as p1 to the power let's say m1 into p2 to the power m2 into p3 to the power m3 into so on and so forth and let's say pn to the power mn okay. So where p1, p2 where where p1, p2 and till pn all are unique prime numbers unique prime numbers and m1 m2 m3 m1 comma m2 comma till mn all are positive positive or rather non-negative whichever way you want to call it positive integers. If it is non non-negative then zero is also included then every prime number will be there in the expression of a. So we will restrict to positive integers fair enough. Now so how do we use this term to prove this particular question? So we say that let us and we will be using something called the method of contradiction. So we say that let let us say that there is a highest prime number so let us say p1 let us say I will write it like this so the let us say let us say that pn is the highest highest prime number okay. So the question is saying that there is not it's not possible that there is a highest prime number but then for you know what did we say we say that we'll be using the method of contradiction and we are saying let us say that pn is the highest prime number okay very good. If pn is the highest prime number then all other prime numbers let's say other other prime numbers being so p1 p2 p3 so on and so forth till pn are the only possible only possible prime numbers now prime numbers so p1 is the first prime number you can say p1 is equal to 2 p2 is equal to 3 and so on and so forth and pn whatever pn value is is the highest possible prime number now we can always define a number let's say q q is equal to let's let us say we are saying q is equal to p1 into p2 into p3 into so on and so forth till pn okay it's highly possible to define a number so clearly q is a composite number q is equal to a composite number composite number isn't it so hence q is defined as product of all prime numbers and pn being the highest one okay so this is this is by you know this is clearly another manifestation or example of our fundamental theorem of arithmetic so one composite number q is expressed as product of prime numbers and this product is all the prime numbers involved are unique okay now let us define another number r such that r is equal to q plus 1 r is equal to q plus 1 now using Euclid's division lemma you know if you remember we can say that r is equal to r is equal to p1 times k1 plus 1 isn't it where where this remainder 1 is correct this is by Euclid's division lemma which we had seen in the previous sessions if you have not seen this session then you I would request you to go back in this series and you will find one video on Euclid's division lemma so r is p1 into k1 plus 1 why because p1 divides q if you see p1 here divides q so hence this particular expression can be written as p1 into k1 where k1 is equal to where k1 if you see is equal to nothing but p2 into p3 into all that till pn isn't it and one because one is here so r is equal to q can be written as p1 into k1 and one is here very good similarly r can be written as p2 into k2 plus 1 now similarly r can also be written as p3 into k3 plus 1 where all k1 k2 k3 all are different integers positive integers okay different integers now so similarly going by this trend I can say r can also be written as pn into kn plus 1 where again all k1 k2 k3 dot dot dot till kn all are positive integers positive integers correct so that means from here we can also say that p1 doesn't divide r because remainder is always 1 similarly p2 also doesn't divide r and so on and so forth we can say pn doesn't divide r that means all the existing prime numbers since pn was the highest prime number that means all the existing prime numbers do not divide r there are two possibilities now so if this is the case if this is the case that all the prime numbers are not dividing right not dividing r that means two possibilities two possibilities are there what is possibility a either r itself is a prime number is a prime number right why because if none of the prime numbers which are existent backward that is what we have proved that pn is the highest prime number and none of them is dividing r that means r itself is a prime number or or there exists there exists a prime number prime number p greater than pn which divides which divides r right only two possibilities are there and under both these possibilities under both these possibilities you will see under both these possibilities you will see that under both these possibilities you will see that there exists so if r itself is a prime number and r is r is equal to p1 into p2 plus dot dot dot pn plus 1 so clearly r is greater than pn right so r is a prime number which is greater than pn number that means pn was not the highest highest prime number right this was in case of possibility a in case of possibility b what is possibility b possibility b is there exists a prime number p greater than pn which divides r so let us write it here so in in case of b so there exists p which is greater than pr that means this means this means there exists a prime greater than pr yeah in both the cases there exists a prime number which is greater than pr so hence in this case also pn is not the highest highest prime number prime number so what did we prove we prove that that you cannot say that there is a highest prime number because every time you consider one of the prime numbers so found as highest you will always get a prime number which is more than whatever last prime number has been found out I hope you understood the question and the solution so let us take another such problem in the next video thank you