# Primes **URL:** \ **Category:** Factual Questions\ **Created:** [October 30, 2014, 12:54pm UTC](https://boards.straightdope.com/t/primes/702997 "2014-10-30T12:54:11Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 30, 2014, 7:09pm UTC](https://boards.straightdope.com/t/primes/702997/41 "2014-10-30T19:09:46Z")
> [@Great\_Antibob](#): > > Again, this is true with caveats. For any given N, primes less than N of the form 6n-1 will outnumber primes of the form 6n+1 for most choices of N. But not for all choices of N. > > The populations do “flip” for certain values of N. I’ve yet to find this value of N where the flip occurs if anyone finds it I humbly admit I was wrong. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [October 30, 2014, 7:25pm UTC](https://boards.straightdope.com/t/primes/702997/42 "2014-10-30T19:25:38Z")
> [@armstrongm](#): > > I’ve yet to find this value of N where the flip occurs if anyone finds it I humbly admit I was wrong. Might take a while. The first flip takes place nearly at 609 Billion. The value was discovered in 1976. **Indistinguishable** gave the exact value back in [post 28](http://boards.straightdope.com/sdmb/showpost.php?p=17859376&postcount=28): > [@Indistinguishable](#): > > That having been said, there are also infinitely many size limits for which this is false (accounting for the other 0.10% of the time). The first example is at size limit 608981813029. Here’s [an article](http://arxiv.org/pdf/math/0408319v1.pdf) that gives plenty more detail. Note that they use 3k+2 and 3k+1 instead, which is equivalent to using 6k-1 and 6k+1. So, yes, for “most” N (defining “most” as we have above), the pattern holds. But also as noted, it fails to hold for some values (one of which given above) and does so for infinitely many such values. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 30, 2014, 7:31pm UTC](https://boards.straightdope.com/t/primes/702997/43 "2014-10-30T19:31:38Z")
> [@Great\_Antibob](#): > > Here’s [an article](http://arxiv.org/pdf/math/0408319v1.pdf) that gives plenty more detail. Note that they use 3k+2 and 3k+1 instead, which is equivalent to using 6k-1 and 6k+1. Right, although, I realized that I failed to properly correct for the behavior of 2 being slightly different between them. Still, among the primes less than or equal to 608,981,813,029, there are 2 more of the form 6n + 1 than of the form 6n - 1, so the first crossing over point is prior to this, and I’d be surprised if it was much earlier. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 30, 2014, 9:07pm UTC](https://boards.straightdope.com/t/primes/702997/44 "2014-10-30T21:07:57Z")
> [@Great\_Antibob](#): > > Might take a while. The first flip takes place nearly at 609 Billion. The value was discovered in 1976. > > **Indistinguishable** gave the exact value back in [post 28](http://boards.straightdope.com/sdmb/showpost.php?p=17859376&postcount=28): > > Here’s [an article](http://arxiv.org/pdf/math/0408319v1.pdf) that gives plenty more detail. Note that they use 3k+2 and 3k+1 instead, which is equivalent to using 6k-1 and 6k+1. > > So, yes, for “most” N (defining “most” as we have above), the pattern holds. But also as noted, it fails to hold for some values (one of which given above) and does so for infinitely many such values. Oh I didn’t understand that’s what that number represented I am sorry for the misunderstanding. Thank you very much for your help. ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [October 30, 2014, 9:43pm UTC](https://boards.straightdope.com/t/primes/702997/45 "2014-10-30T21:43:14Z")
Is this a better way of asking the question: Given a random integer, what is the probability that the number of 6n-1 primes (less than that number) is greater than the number of 6n+1 primes? The assertion is that this probability is greater than .5. Stated as a probability, it should be well-defined, and sidestep the issue of counting infinite sets. My guess is that it’s equivalent to the density interpretation, but I don’t pretend to understand the density interpretation. ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [October 30, 2014, 10:03pm UTC](https://boards.straightdope.com/t/primes/702997/46 "2014-10-30T22:03:20Z")
First, I see that my post above about probability is old news already. I do think it’s the better way to think of the problem. > [@armstrongm](#): > > The point is that I am thoroughly convinced that at infinity they will not be the same size. I think you have a deep misunderstanding of the nature of the different infinite cardinalities. It’s a really confusing subject, and I’m far from an expert, but I can point out a couple things. First, “infinity divided by two is infinity.” That is, the cardinality of the integers is EXACTLY EQUAL to the cardinality of the odd integers. That seems odd, since the odd set is missing half of the integers! But it’s true by the definition of cardinality that makes sense for infinite sets. As mentioned above, the way we work out cardinalities for a given infinite set is by doing a one-to-one-and-onto mapping between the set and some other known set (such as the integers, or ordinals, or cardinals.) It’s easy to map from integers to odds, and it’s clear that when we map from integers to odds, we hit every odd. And we can do this the other way too (with a different mapping). So, the two sets have the same cardinality. They’re the same size. If you pick a definition for cardinality of infinites that doesn’t maintain this property, you either don’t get very far, or you bump into contradictions. The two infinite cardinalities that I’m vaguely aware of are: The number of integers (aleph-null) The number of points on a line (not sure what this is named) There’s a “continuum hypothesis” that posits that there’s an infinite number of infinite cardinalities between these two (or something like that) but it’s a bit over my beanie. Now, just to confuse us all, let’s do an experiment. Two, actually, each involving a rather large hat, into which we drop slips of paper, once every second. Hat 1: Every second, we put in 10 slips of unmarked paper and take out 9. Hat 2: Every second, we put in 10 numbered slips of paper (starting with 1 through 10) and take out 1 (starting with 1). At the end of eternity, how many slips of paper are in each hat? Hat 1: infinitely many Hat 2: none For Hat 2, we know the hat is empty because for any given slip of paper, we can tell you exactly when we took it out. I’m not sure how we know the hat has infinitely many for Hat 1, but I trust Martin Garner or whomever I got this tidbit from. 😉 Infinity and intuition don’t mix. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 31, 2014, 8:15am UTC](https://boards.straightdope.com/t/primes/702997/47 "2014-10-31T08:15:13Z")
> [@Learjeff](#): > > First, I see that my post above about probability is old news already. I do think it’s the better way to think of the problem. > > I think you have a deep misunderstanding of the nature of the different infinite cardinalities. It’s a really confusing subject, and I’m far from an expert, but I can point out a couple things. > > First, “infinity divided by two is infinity.” That is, the cardinality of the integers is EXACTLY EQUAL to the cardinality of the odd integers. That seems odd, since the odd set is missing half of the integers! But it’s true by the definition of cardinality that makes sense for infinite sets. > > As mentioned above, the way we work out cardinalities for a given infinite set is by doing a one-to-one-and-onto mapping between the set and some other known set (such as the integers, or ordinals, or cardinals.) It’s easy to map from integers to odds, and it’s clear that when we map from integers to odds, we hit every odd. And we can do this the other way too (with a different mapping). So, the two sets have the same cardinality. They’re the same size. If you pick a definition for cardinality of infinites that doesn’t maintain this property, you either don’t get very far, or you bump into contradictions. > > The two infinite cardinalities that I’m vaguely aware of are: > > The number of integers (aleph-null) > The number of points on a line (not sure what this is named) > > There’s a “continuum hypothesis” that posits that there’s an infinite number of infinite cardinalities between these two (or something like that) but it’s a bit over my beanie. > > Now, just to confuse us all, let’s do an experiment. Two, actually, each involving a rather large hat, into which we drop slips of paper, once every second. > > Hat 1: Every second, we put in 10 slips of unmarked paper and take out 9. > Hat 2: Every second, we put in 10 numbered slips of paper (starting with 1 through 10) and take out 1 (starting with 1). > > At the end of eternity, how many slips of paper are in each hat? > > Hat 1: infinitely many > Hat 2: none > > For Hat 2, we know the hat is empty because for any given slip of paper, we can tell you exactly when we took it out. > > I’m not sure how we know the hat has infinitely many for Hat 1, but I trust Martin Garner or whomever I got this tidbit from. 😉 > > Infinity and intuition don’t mix. Thank you very much. I thought that as you get different levels of infinity through cardinality. If you work on a 3D plan then, this is more a memory exercise as I read it somewhere, you get some infinities that are larger than others. So my initial thought was that one could then not say that infinity/2 was equal to infinity I thought that rule only applied to addition. Thank you very much for you explanation on the subject. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 31, 2014, 8:20am UTC](https://boards.straightdope.com/t/primes/702997/48 "2014-10-31T08:20:31Z")
> [@Learjeff](#): > > Is this a better way of asking the question: > > Given a random integer, what is the probability that the number of 6n-1 primes (less than that number) is greater than the number of 6n+1 primes? The assertion is that this probability is greater than .5. > > Stated as a probability, it should be well-defined, and sidestep the issue of counting infinite sets. My guess is that it’s equivalent to the density interpretation, but I don’t pretend to understand the density interpretation. Let me just correct you before I answer to the best of my knowledge primes only apply to natural numbers. Small technicality though. As far as I understand it the density this. Let’s say, for the sake of argument, that in the first 100 numbers there are 25 primes. The density is 0.25. But for the first thousand there are 200 primes. Therefore the density of primes for the first 1000 is 0.2. The best way to state it is this I think, "If you take all the primes until infinity what is the probability that primes can be written as 6n-1 and what is the probability that they can be written as 6n+1. Hence the problem of infinite sets. That is unless you have a better idea? I am all hears. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 31, 2014, 8:26am UTC](https://boards.straightdope.com/t/primes/702997/49 "2014-10-31T08:26:42Z")
If anyone is feeling brave here is another problem that has been bugging me for a while. I’m not going to explain the whole process and the proof is long. Let’s say that you could divide prime differences into three sets. The first set is when you divide the prime difference and the result is an odd number. The second is when you divide the prime difference and the result in an even number. The third is when the differences between primes which are multiples of 6. The million dollar question now is this. What is the probability and frequency of each set for this prime difference? ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 31, 2014, 8:40am UTC](https://boards.straightdope.com/t/primes/702997/50 "2014-10-31T08:40:32Z")
As a final question if anyone would be so kind as to check my proof before I submit it I would be very grateful. I would happily include you name in the list of authors before I sent it in. The maths is sound I just want to check that the layout and everything makes sense ---
**Author:** ![EllisDee](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ellisdee/32/4531_2.png) [@EllisDee](https://boards.straightdope.com/u/EllisDee)\ **Post date:** [October 31, 2014, 9:34am UTC](https://boards.straightdope.com/t/primes/702997/51 "2014-10-31T09:34:56Z")
> [@armstrongm](#): > > The point is that I am thoroughly convinced that at infinity they will not be the same size. It is so unlikely. From what I understand there is a 0.1% chance of it happening. NASA got man to the moon on a 99.9% chance. Through chebyschev’s bias it must be the case that for values there are more primes that are written as 6n-1. You keep saying at infinity they will be equal. But that’s a bit like saying when we reach the each of the universe this happens. It’s so far beyond any number you can think of that it seems almost pointless. > > I’m not attacking you at all its very interesting. But I cant see it happening I have yet to hear an argument that doesn’t support the idea that at infinity even or at the very least incredibly large values there aren’t primes more primes than can be written as 6n-1 than 6n+1. I’m having PTSD flashbacks to the very old SDMB thread about whether or not 0.999… = 1. They are equal. 0.999… is exactly equal to 1. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [October 31, 2014, 9:41am UTC](https://boards.straightdope.com/t/primes/702997/52 "2014-10-31T09:41:38Z")
not the same thing the reason that when you calculate it you get 0.999… as being equalt to one is that there are no numbers between them ---
**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\ **Post date:** [October 31, 2014, 12:12pm UTC](https://boards.straightdope.com/t/primes/702997/53 "2014-10-31T12:12:40Z")
> [@armstrongm](#): > > The best way to state it is this I think, "If you take all the primes until infinity what is the probability that primes can be written as 6n-1 and what is the probability that they can be written as 6n+1. Hence the problem of infinite sets. This would be said a little more precisely as: A = limit as n -\> infinity { Pi[sub]-/sub / Pi(n) } and B = limit as n -\> infinity { Pi[sub]+/sub / Pi(n) }, where Pi(m) is the number of primes less than or equal to m, Pi[sub]-/sub is the number of primes of the form 6n-1 less than or equal to m, and Pi[sub]+/sub is the number of primes of the form 6n+1 less than or equal to m. **That Don Guy** got values of 0.999935 and 0.999916 for A/B, for n = 500 million and 1 billion, respectively, for that calculation. Ignoring the primes 2 and 3, A+B = 1, so you can work out A and B. I don’t think anyone has given the limit as n–\>infinity for these values. Note that this is not the same calculation as the one **Indistinguishable** described, which has a value of about 0.9990 in the limit. They may be related, but I don’t think there’s any reaosn to think they are equal. ---
**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\ **Post date:** [October 31, 2014, 12:42pm UTC](https://boards.straightdope.com/t/primes/702997/54 "2014-10-31T12:42:29Z")
There is a theorem that covers this. To state it in a general form, let N \> 2 be a fixed integer. Suppose k\_1 = 1, k\_2, …, k\_n = -1 be the set of numbers \< N and having no common divisor with N. The number of them is called phi(N) and is easy to calculate if you know the prime decomposition of N. Obviously phi(6) = 2. Now let P\_1, …, P\_n denote the set of primes that have remainders of k\_1, …, k\_n, respectively, when divided by N. Then for sufficiently large M, the number of primes less than M in each of the classes P\_1, …, P\_n is arbitrarily close to pi(M)/n, where pi(M) is the number of primes less than M. In other words, all these classes have. asymptotically, the same number of elements. In the case at hand, ratio of the two sets approaches 1. This does not preclude the possibility that the difference grows large. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 31, 2014, 5:48pm UTC](https://boards.straightdope.com/t/primes/702997/55 "2014-10-31T17:48:02Z")
> [@ZenBeam](#): > > This would be said a little more precisely as: > A = limit as n -\> infinity { Pi[sub]-/sub / Pi(n) } and B = limit as n -\> infinity { Pi[sub]+/sub / Pi(n) }, where Pi(m) is the number of primes less than or equal to m, Pi[sub]-/sub is the number of primes of the form 6n-1 less than or equal to m, and Pi[sub]+/sub is the number of primes of the form 6n+1 less than or equal to m. **That Don Guy** got values of 0.999935 and 0.999916 for A/B, for n = 500 million and 1 billion, respectively, for that calculation. Ignoring the primes 2 and 3, A+B = 1, so you can work out A and B. I don’t think anyone has given the limit as n–\>infinity for these values. > > Note that this is not the same calculation as the one **Indistinguishable** described, which has a value of about 0.9990 in the limit. They may be related, but I don’t think there’s any reaosn to think they are equal. Sorry, you asked about this earlier but I’ll respond to it now. A = B = 1/2, so A/B = 1; this is the Dirichlet result that has been mentioned many times. That is, the ratio Pi[sub]-/sub/Pi[sub]+/sub tends to 1. But for some n this ratio is above 1 and for some n this ratio is below 1. This is what the calculation I noted addresses: in a certain sense, there is a precise percentage L of n such that this ratio is above 1, and this L is specifically between 99.90% and 99.91%. I might elaborate a little on “in a certain sense”. Let f(n) be 1 or 0 according as to whether the ratio Pi[sub]-/sub/Pi[sub]+/sub is above 1 or not. We might ask “What is the limiting value of (the average of f(n) over the range up through m) as m grows large?”. Now, using the ordinary notion of “average”, we get such oscillations as that this limit actually fails to exist. But we can take a weighted average instead (in such a way as that, were the ordinary limiting average to exist, it would have to equal the weighted limiting average), and in so doing, we get the L noted above. The specific weighting we use corresponds to what is called “logarithmic”, as opposed to “uniform” density; it’s as though, instead of taking all the values up through m and averaging f over them in the ordinary way, instead, we first group values by their number of digits, then take the average of f within each of these groups, then average those averages together WITHOUT accounting for the fact that groups with more digits also have many more members. ---
**Author:** ![Stealth\_Potato](https://avatars.discourse-cdn.com/v4/letter/s/d78d45/32.png) [@Stealth\_Potato](https://boards.straightdope.com/u/Stealth_Potato)\ **Post date:** [October 31, 2014, 6:17pm UTC](https://boards.straightdope.com/t/primes/702997/56 "2014-10-31T18:17:56Z")
> [@EllisDee](#): > > I’m having PTSD flashbacks to the very old SDMB thread about whether or not 0.999… = 1. > > They are equal. 0.999… is exactly equal to 1. No! No! Don’t mention that thread! You might summon it again from the outer darkness! ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [October 31, 2014, 6:23pm UTC](https://boards.straightdope.com/t/primes/702997/57 "2014-10-31T18:23:29Z")
> [@Exapno\_Mapcase](#): > > I’m looking at the question in the OP. “But can more primes be expressed as 6n+1 than 6n-1 or is the number of primes that can be expressed as 6n+1 equal to the number of primes that can be expressed as 6n-1?” > > I’m not saying that the differences at any point are not interesting. I’m saying that those are a different question from the one I’m reading. And also a different one than the OP who says that density is not his interest is asking. And one that Great Antibob seemed to answer, giving the same answer as mine with a reference to a proven theorem. (Yes, redundant technically but not in colloquial English.) > > Could you please clarify? I’m seriously missing your point. Sure. And I apologize for any harshness in my tone. It seemed, from reading **armstrongm** ’s posts, that they were specifically interested in the question of whether, up to any particular finite limit, there were always more primes of the form 6n - 1 than of the form 6n + 1. This is clearest in their second post in the thread. They phrased this at times as the question of whether there were more primes of the one form than the other, without explicitly mentioning what was meant by more primes of the one form than the other. For example, in the OP. But no matter; later context revealed what they meant by this. (They also explicitly said they were not interested in density, which is in fact the right thing to say, because this question about whether there are always more of the one form than the other is not affected by the fact that the two have equal density) Because of the OP, however, others (never **armstrongm** themself) introduced into the thread the idea of cardinality and the observation that the two forms gave rise to sets of equal cardinality. Which is true, but irrelevant to the notion of “more” that **armstrongm** was interested in. And I was distressed to see that **armstrongm** seemed to be getting flak simply for expressing their interest in studying a different notion of “more” than the one others were noting. I think those of little mathematical sophistication are often unduly pummeled for not expressing themselves in orthodox mathematical language, even when the concepts they are grasping for (however inchoately) are perfectly sensible (and the language being used by the amateur to describe it is perfectly reasonable, and indeed is drawn from a perspective with which it highlights important analogies, but just happens not to match some arbitrary (and often relatively recent) tradition of how mathematicians speak). And so I am perhaps quick to “defend” them. Finally, I will note that, though **armstrongm** apparently has a proof they seek to publish, which is very likely to eventually take this thread into crank-territory, all the above still applies. Whatever further crankery comes in, **armstrongm** did make a legitimate observation and have a legitimate question, which had nothing to do with cardinality even though they used the language of “more” and so on, and I think it was basically fine of them to ask about the things they were interested in using the language that they did. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [November 1, 2014, 12:38am UTC](https://boards.straightdope.com/t/primes/702997/58 "2014-11-01T00:38:43Z")
The one thing that frustrates me about primes is, whenever you come up with a rule about primes, you will inevitably find exceptions to the rule. And when you come up with a rule to explain the exceptions, there will be exceptions to _that_ rule. All primes are of the form 6x+/- 1…except for 2 and 3… …and where 6x+/-1=(6x+/-1)(6x+/-1), such as 25,35,49… so x itself must be some multiple of 6…except when it doesn’t, such as 143 and 145…:mad: ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 1, 2014, 12:50am UTC](https://boards.straightdope.com/t/primes/702997/59 "2014-11-01T00:50:29Z")
The rule “All primes are of the form 6x+/- 1…except for 2 and 3…” has no further exceptions. It doesn’t claim the converse “All numbers of the form 6x+/-1 are prime”, so 25, 35, 49, etc., are no problem for it. And I don’t see why you think x itself must be some multiple of 6 [x can perfectly well be 1 (as in 5 and 7) or 2 (as in 11 and 13) or 3 (as in 17 and 19) or 7 (as in 41 and 43) or a million other things which aren’t multiples of 6]. ---
**Author:** ![armstrongm](https://avatars.discourse-cdn.com/v4/letter/a/50afbb/32.png) [@armstrongm](https://boards.straightdope.com/u/armstrongm)\ **Post date:** [November 4, 2014, 10:48am UTC](https://boards.straightdope.com/t/primes/702997/60 "2014-11-04T10:48:10Z")
> [@Enola\_Straight](#): > > The one thing that frustrates me about primes is, whenever you come up with a rule about primes, you will inevitably find exceptions to the rule. > > And when you come up with a rule to explain the exceptions, there will be exceptions to _that_ rule. > > All primes are of the form 6x+/- 1…except for 2 and 3… > > …and where 6x+/-1=(6x+/-1)(6x+/-1), such as 25,35,49… > > so x itself must be some multiple of 6…except when it doesn’t, such as 143 and 145…:mad: no x doesn’t have to be a multiple of 6. In fact x can be any natural number. [Previous page](https://boards.straightdope.com/t/primes/702997.md?page=2) [Next page](https://boards.straightdope.com/t/primes/702997.md?page=4)