Can you prove that there are arbitrarily many primes in arbitrarily big intervals?<\/p>\n\n
The problem statement asks you to prove that for any<\/em> positive integer number \\(k\\)<\/span> you can think of,\nthere will be a certain lower bound \\(n_0\\)<\/span> (that depends on \\(k\\)<\/span>) such that for any<\/em> integer \\(n \\geq n_0\\)<\/span>,\nthere is<\/em> an interval of length \\(n\\)<\/span> that contains exactly<\/em> \\(k\\)<\/span> primes.\n(When we talk about the length of the interval, we are talking about how many integers it contains.)<\/p>\n If the statement confused you a bit, that is ok.\nLet me rephrase it.<\/p>\n You and I will play a game.\nI will think of a positive integer \\(k\\)<\/span>.\nNow, your job is to come up with another positive integer \\(n_0\\)<\/span> such that,\nif I pick a number \\(n\\)<\/span> greater than \\(n_0\\)<\/span>,\nyou can always find an interval of size \\(n\\)<\/span> that contains \\(k\\)<\/span> primes.<\/p>\n For example, if I thought of \\(k = 5\\)<\/span>, you could not<\/em> pick \\(n_0 = 4\\)<\/span>.\nWhy not?\nBecause if I pick \\(n = 4\\)<\/span>, there is no<\/em> interval of length \\(4\\)<\/span> that contains \\(5\\)<\/span> prime numbers...\nEspecially because an interval of length \\(4\\)<\/span> contains only \\(4\\)<\/span> integers!<\/p>\n This problem was posed to me by my mathematician cousin and I confess that worried me a bit.\nFunnily enough, the problem has a surprisingly simple solution.\n(I am not saying you will get there easily.\nI am just saying that once you do, you will realise the solution was not very complicated.)<\/p>\n Give it some thought!<\/p>\n<\/div>\n Remember<\/strong>:<\/p>\n Congratulations 🎉 to everyone who managed to solve this problem:\nCongratulations to you if you managed to solve this problem correctly!\nIf you did, feel free to<\/p>\n If you<\/em> managed to solve this problem, you can add your name to the list<\/a>!\nYou can also email me<\/a> your solution and we can discuss it.<\/p>\n A thing I like about this problem is that not only can you prove that interesting statement about the prime numbers, but you can also determine exactly what the lower bound \\(n_0\\)<\/span> is.<\/p>\n Let us say that \\(p_k\\)<\/span> is the \\(k\\)<\/span>-th prime.\nThen, if we set \\(n_0 = p_k\\)<\/span>, we are good to go.\nLet me show you why.<\/p>\n Suppose that \\(n\\)<\/span> is any integer \\(n \\geq n_0\\)<\/span>.\nThen, the interval \\([1, n]\\)<\/span> contains \\(p_k\\)<\/span> in it.\nWhy?\nBecause \\(p_k = n_0\\)<\/span> and \\(n \\geq n_0\\)<\/span>.<\/p>\n So, the interval \\([1, n]\\)<\/span> contains \\(k\\)<\/span> or more<\/em> prime numbers.\nIf it contains \\(k\\)<\/span> prime numbers, we just found our interval of length \\(n\\)<\/span> that contains exactly \\(k\\)<\/span> primes.\nIf it contains more than \\(k\\)<\/span> primes, we must do something about it.<\/p>\n If the interval \\([1, n]\\)<\/span> contains more than \\(k\\)<\/span> prime numbers, then we start sliding the interval to the right, like so:...<\/p>","summary":"Can you prove that there are arbitrarily many primes in arbitrarily big intervals? In this mathematical problem, which is just a logic challenge, the solution is surprisingly uncomplicated.","date_modified":"2025-07-23T16:49:02+02:00","tags":["logic","mathematics","number theory","primes"],"image":"\/user\/pages\/02.blog\/03.problems\/p063-arbitrarily-many-primes-in-arbitrarily-big-intervals\/thumbnail.webp"},{"title":"Twitter proof: consecutive integers are coprime","date_published":"2020-11-14T00:00:00+01:00","id":"https:\/\/mathspp.com\/blog\/consecutive-integers-are-coprime","url":"https:\/\/mathspp.com\/blog\/consecutive-integers-are-coprime","content_html":" Let's prove that if \\(k\\)<\/span> is an integer, then \\(\\gcd(k, k+1) = 1\\)<\/span>. That is, any two consecutive integers are coprime.<\/p>\n\nSolvers<\/a><\/h2>\n
Solution<\/a><\/h2>\n