You Can Build a Million-Number Gap With No Primes

[00:00:00]Here are the primes laid out on a line.

[00:00:03]2, 3, 5, 7, 11, 13. Down at the start, they crowd together, almost elbow to elbow. But, watch what happens as you walk to the right. The dots begin to thin. The empty stretches between them get longer. Between one prime and the next, there is a gap, a run of numbers [music] with nothing prime inside it.

[00:00:24]Near the beginning, those gaps are tiny, just one or two steps. But, by the time you reach 100, 13, the next prime is all the way out at 127.

[00:00:34]That is 13 numbers in a row without a single prime among them.

[00:00:38]So, here is the question that runs underneath this whole video. How big can one of these gaps get? Is there some ceiling, some longest possible run of prime-free numbers that the primes [music] will never let you exceed? Or, can the empty stretches grow without any limit at all?

[00:00:55]The answer turns out to be strange, [music] and you can hold both halves of it in your hand at once. Let us just measure the gaps as we go, and keep a record book. Every time we find a gap longer than anything we have seen before, we write it down along with the prime it starts from.

[00:01:11]The first real gap of four opens up right after seven. A gap of six shows up after 23. After 89, we get our first stretch of eight prime-free numbers.

[00:01:21]Then a jump to 14 after 113, 18 after 523, 20 after 887.

[00:01:29]And then something that feels almost sudden. Just after 1,327, the primes go quiet for 34 numbers in a row. That is the first gap that really surprises you. Everything before it grew slowly, step by step. And then this one nearly doubles the record in a single jump. Keep scanning, and the records keep falling. 36, 44, 52, 72. The empty stretches just keep setting new highs.

[00:01:58]There is no sign of them settling down.

[00:02:00]Now, instead of hunting for gaps, let us build one. Pick any length you want, and I will hand you a run of that many consecutive numbers with no primes inside on demand. Say you want a desert of length four, four numbers in a row all composite. Here is the recipe. Take five factorial, which means 5 * 4 * 3 * 2 * [music] 1. That equals 120.

[00:02:26]Now, write down the four numbers 122, 123, 124, 125.

[00:02:33]Every one of those four is composite, and you can see exactly why. 122 is even. 123 is 3 * 41. 124 is 4 * [music] 31. 125 is 5 * 25. Four numbers in a row, >> [music] >> not a prime among them, and we built it deliberately.

[00:02:54]The same trick gives a desert of length five from six factorial, or length six from seven factorial.

[00:03:01]Whatever length you name, there is a starting point waiting. The reason this always works is one line, and once you see it, you will never forget it. Take the length you want, call it L. Look at the factorial of L + 1. That factorial is the product of every whole number from two up to L + 1. So, every one of those numbers divides it evenly. Now, add two to the factorial, then three, then four, all the way up to L + 1. That is your run of numbers.

[00:03:31]>> [music] >> Take any one of them, the factorial plus some number K. The factorial is divisible by K because K was one of the things multiplied in, and K is obviously divisible by K. So, their sum is divisible by K as well. Every number in the run has a divisor sitting right there in plain sight, which means every number in the run is composite. That is the whole proof. So, if you want a desert of a million numbers, you take 1,000,001 factorial and add 2, 3, 4 on up to 1,000,001.

[00:04:03]A million consecutive composites guaranteed, no searching required. There is a catch, and it is a funny one.

[00:04:10]This construction proves that deserts of every length exist, but it drops them absurdly far out on the number line.

[00:04:18]Think about that million number desert we just built. It starts at 1,000,001 factorial.

[00:04:24]That number is not big in any ordinary sense. Written out in full, it has more than 5 million digits.

[00:04:31]You could not fit it on this screen, or in this whole video, or in a stack of books.

[00:04:37]The construction is honest, but it is wildly wasteful.

[00:04:40]Because here is the thing, the real gaps show up much, much earlier. Remember that surprising gap of 34? It happens right after 1,327, a number with four [music] digits. The factorial recipe asked for a desert that long would place it out past a number with dozens of digits.

[00:05:00]So, the construction is a proof, not a map. It shows you that long deserts must exist, but the actual ones are hiding far closer to home than it would ever suggest. If you step back from the records and just ask about a typical gap, there is a clean answer, and it comes from the deepest result about how primes are spread out.

[00:05:20]The primes thin out at a very definite rate. Around a number X, the average gap between consecutive primes is [music] close to the natural logarithm of X.

[00:05:29]Near 1,000, the average gap is about seven. Near a million, it is about 14.

[00:05:35]Near a billion, about 21. So, the deserts, on average, keep getting longer the further out you go. And I checked this against real primes, not just the formula.

[00:05:45]In a window of 40,000 numbers sitting right around a million, the actual average gap came out to 13.8, almost exactly the logarithm of a million.

[00:05:55]The formula and the real primes agree.

[00:05:56][music] This is the slow, steady side of the story. The average gap grows without bound, but it grows gently, like the logarithm. One extra step every time you multiply the size of your numbers by E.

[00:06:09]Now for the twist, and this is the part worth carrying home.

[00:06:13]The gaps have no ceiling. We have proven that deserts of any length exist. And yet, infinitely often, the gaps stay small. Both things are true at the same time.

[00:06:25]Here is what that means. You can find prime-free stretches as long as you like, a thousand, a million, [music] any number at all. But no matter how far out you go, primes keep coming back close together, too. In 2013, a mathematician named Yitang Zhang proved that infinitely many pairs of primes sit within 70 million of each other. That was the first fixed bound [music] anyone had ever found. Within months, the number came crashing down.

[00:06:54]James Maynard [music] sharpened the method to 600. A large collaboration grounded further, all the way to 246.

[00:07:03]So, it is now a proven fact that infinitely often, two primes come within 246 of each other. No ceiling on how big a gap can get, but a proven floor that the primes return to again and again.

[00:07:16]The deserts and the crowds coexist forever. So, the empty stretches grow without limit, but how fast does the biggest possible gap grow?

[00:07:26]That question is still open, and the best guess is beautiful. The average gap near P is the logarithm of P.

[00:07:33]But the biggest gaps, [music] the record-setting deserts, run larger than average. In the 1930s, Harald Cramér conjectured that the largest gap you will find [music] near a prime P is about the logarithm of P squared, not the logarithm, but it's square. [music] And the real records fit this shape surprisingly well. Near 1,327, the logarithm squared is about 52, and the actual record gap there was 34, the same ballpark.

[00:08:03]Near 31,000, the guess is around 107, but this is only a conjecture. Nobody has proven how fast the maximal gaps truly [music] grow.

[00:08:14]We know the deserts get longer forever, and we can build one of any size on command. Yet, the exact law for the largest of them remains one of the open questions about the primes. So, here is the whole shape of it. The primes never stop. Euclid proved that more than 2,000 [music] years ago, and nothing has changed it.

[00:08:33]And yet, they fall silent for [music] stretches as long as you please. You can order a desert of any length, 100, 1,000, a million consecutive numbers with no [music] primes inside, and the recipe fits on one line.

[00:08:46]Take the factorial of one more than the length, and add two, three, four on up.

[00:08:51]Every term [music] has a small divisor hidden in it, so every term is composite. That is the tension worth keeping. The primes are infinite, guaranteed [music] to go on forever, and at the very same time, they leave silences of every length behind them. No ceiling on the deserts, a floor of 246 that they keep returning to, and a construction so simple, you could hand someone a million number gap on the back of a napkin.

[00:09:17]The primes never end, but they leave room, as much room as you could ever ask for. This was one more in a series on the questions the primes refuse to settle quietly. Stay for the rest.