# Is it possible to find EXACTLY $101$ consecutive composite numbers

Here is a similar question that asks for $$101$$ numbers none of them are prime and it is well known that $$101!+1,101!+2,101!+3,\cdots,101!+101$$ are those numbers. I am interested to know, how we can find exactly $$101$$(more generally, $$n$$) consecutive composites, i.e; if the list of composites is $$k,k+1,\cdots,k+100$$, then neither $$k-1$$ nor $$k+101$$ are composite,$$($$i.e; $$k-1,k+101\in \mathbb{P})$$.

From this OEIS list, and from wikipedia also, $$n$$ for which we have primes of the form $$n!+1$$ are listed. This list doesn't includes $$101$$, that means, $$101!+1$$ is composite also. So $$101!+1,101!+2,101!+3,\cdots,101!+101$$ is not list of exactly $$101$$ consecutive composite numbers(as, $$101!+1$$ is also composite). Is there a way to find such list for $$101$$ consecutive composites? and is it possible to find such list for any given length $$n\in\mathbb{N}$$ which includes exactly $$n$$ consecutive composite numbers.

I have tried this for a while and not getting any way out. In other way it is similar to finding primes with gap $$102$$. Though it is not listed here.

Edit:

I don't think there can be any solution without programming. I have translated this code to Python, one can check here.

• Higher up all the primes are odd numbers meaning that there is always an odd number of composites between any two consecutive primes. Therefore exactly 100 is impossible. Aug 7, 2018 at 8:20
• Not an efective method, but a version of Polignac's conjecture asserts that there are infinitely many such primes $p$ such the next prime is $p+100$ (or $p+n$ for every even number). Aug 7, 2018 at 8:21
• @JyrkiLahtonen Yeah, but what about odd gap ? suppose I change $100$ to $101$ ? Aug 7, 2018 at 8:23
• Sorry about making a point by stating the obvious. Well, we have users who would have jumped to post that as an answer :-/. No, your real question is probably very difficult in general. I do guess that somebody with a suitable CAS and a bit of spare time could find a prime gap of exactly 100 or exactly 102 in a reasonable time (exactly how long it would take I dare not guess) by searching in a suitable range (primality testing is efficient in a predicted range here). Aug 7, 2018 at 8:28
• @tarit goswami Consider the consecutive primes $7$ and $11$. The gap is filled by $8,9,10$ which is $3$ numbers; odd. See comment by Jyrki Lahtonen. Odd gaps are the only possibility, so gap of exactly $100$ is not possible. Sep 6, 2018 at 1:34

Average gap between primes is ~100 around $e^{100}=2.7\times10^{43}$, but it's amazing how frequently we find such gap for much, much smaller numbers:

Gaps of size 100 or 102:

396733 -> 396833: gap = 100
838249 -> 838349: gap = 100
1313467 -> 1313567: gap = 100
1444309 -> 1444411: gap = 102
1648081 -> 1648181: gap = 100
1655707 -> 1655807: gap = 100
1761187 -> 1761289: gap = 102
1775069 -> 1775171: gap = 102
2242549 -> 2242651: gap = 102
2305169 -> 2305271: gap = 102
2345989 -> 2346089: gap = 100
2784373 -> 2784473: gap = 100
3254959 -> 3255059: gap = 100
3392341 -> 3392443: gap = 102
3595489 -> 3595589: gap = 100
4044077 -> 4044179: gap = 102
4047157 -> 4047257: gap = 100
4315607 -> 4315709: gap = 102
4359403 -> 4359503: gap = 100
4447321 -> 4447423: gap = 102
4535717 -> 4535819: gap = 102
4571107 -> 4571207: gap = 100
4596731 -> 4596833: gap = 102
4665553 -> 4665653: gap = 100
4686709 -> 4686811: gap = 102
4783873 -> 4783973: gap = 100
5209177 -> 5209279: gap = 102
5211109 -> 5211209: gap = 100
5323949 -> 5324051: gap = 102
5398597 -> 5398697: gap = 100
5528287 -> 5528387: gap = 100
5723899 -> 5723999: gap = 100
5837399 -> 5837501: gap = 102
5976079 -> 5976181: gap = 102
6027283 -> 6027383: gap = 100
6242263 -> 6242363: gap = 100
6429223 -> 6429323: gap = 100
6836867 -> 6836969: gap = 102
6845417 -> 6845519: gap = 102
6851863 -> 6851963: gap = 100
7259167 -> 7259267: gap = 100
7309427 -> 7309529: gap = 102
7554367 -> 7554467: gap = 100
7662517 -> 7662617: gap = 100
7683131 -> 7683233: gap = 102
8073647 -> 8073749: gap = 102
8166107 -> 8166209: gap = 102
8257057 -> 8257157: gap = 100
8294021 -> 8294123: gap = 102
8332427 -> 8332529: gap = 102
8350483 -> 8350583: gap = 100
8441869 -> 8441969: gap = 100
8806891 -> 8806991: gap = 100
8841529 -> 8841629: gap = 100
8850671 -> 8850773: gap = 102
8921467 -> 8921569: gap = 102
8939597 -> 8939699: gap = 102
8947217 -> 8947319: gap = 102
8981461 -> 8981563: gap = 102
9162367 -> 9162467: gap = 100
9251059 -> 9251161: gap = 102
9389827 -> 9389927: gap = 100
9413533 -> 9413633: gap = 100
9548999 -> 9549101: gap = 102
9550813 -> 9550913: gap = 100
9573481 -> 9573583: gap = 102
9812611 -> 9812711: gap = 100
9918421 -> 9918521: gap = 100
9964667 -> 9964769: gap = 102
9972811 -> 9972913: gap = 102
10149721 -> 10149823: gap = 102
10172221 -> 10172321: gap = 100
10194227 -> 10194329: gap = 102
10194607 -> 10194707: gap = 100
10295821 -> 10295921: gap = 100
10440641 -> 10440743: gap = 102
10508359 -> 10508461: gap = 102
10569007 -> 10569107: gap = 100
10641937 -> 10642039: gap = 102
10668421 -> 10668521: gap = 100
10670389 -> 10670489: gap = 100
10703741 -> 10703843: gap = 102
10937921 -> 10938023: gap = 102
10942753 -> 10942853: gap = 100
10986277 -> 10986377: gap = 100
11126701 -> 11126803: gap = 102
11330197 -> 11330299: gap = 102
11366947 -> 11367047: gap = 100
11609203 -> 11609303: gap = 100
11765659 -> 11765759: gap = 100
11853031 -> 11853133: gap = 102
11930561 -> 11930663: gap = 102
11936809 -> 11936909: gap = 100
12069517 -> 12069619: gap = 102
12117121 -> 12117221: gap = 100
12459229 -> 12459329: gap = 100
12468457 -> 12468557: gap = 100
12501067 -> 12501169: gap = 102
12522973 -> 12523073: gap = 100
12557317 -> 12557417: gap = 100
12613721 -> 12613823: gap = 102
12623089 -> 12623189: gap = 100
12686647 -> 12686747: gap = 100
12780409 -> 12780511: gap = 102
13161601 -> 13161703: gap = 102
13200001 -> 13200101: gap = 100
13226921 -> 13227023: gap = 102
13251547 -> 13251647: gap = 100
13290547 -> 13290647: gap = 100
13331309 -> 13331411: gap = 102
13339429 -> 13339531: gap = 102
13385431 -> 13385531: gap = 100
13399709 -> 13399811: gap = 102
13582297 -> 13582399: gap = 102
13660663 -> 13660763: gap = 100
13720933 -> 13721033: gap = 100
13734913 -> 13735013: gap = 100
13776067 -> 13776167: gap = 100
13877329 -> 13877431: gap = 102
13905607 -> 13905707: gap = 100
14059657 -> 14059757: gap = 100
14076757 -> 14076857: gap = 100
14159149 -> 14159251: gap = 102
14160877 -> 14160977: gap = 100
14200141 -> 14200243: gap = 102
14243287 -> 14243389: gap = 102
14252431 -> 14252531: gap = 100
14272201 -> 14272301: gap = 100
14292571 -> 14292673: gap = 102
14312281 -> 14312381: gap = 100
14552269 -> 14552371: gap = 102
14933491 -> 14933591: gap = 100
14986009 -> 14986109: gap = 100
15028001 -> 15028103: gap = 102
15073999 -> 15074099: gap = 100
15455851 -> 15455953: gap = 102
15602989 -> 15603089: gap = 100
15666229 -> 15666331: gap = 102
15744541 -> 15744643: gap = 102
15805513 -> 15805613: gap = 100
15817831 -> 15817931: gap = 100
15913399 -> 15913501: gap = 102
16034591 -> 16034693: gap = 102
16117151 -> 16117253: gap = 102
16188649 -> 16188749: gap = 100
16250347 -> 16250447: gap = 100
16652869 -> 16652969: gap = 100
16699901 -> 16700003: gap = 102
16806071 -> 16806173: gap = 102
16988399 -> 16988501: gap = 102
17041789 -> 17041891: gap = 102
17198017 -> 17198119: gap = 102
17208497 -> 17208599: gap = 102
17318023 -> 17318123: gap = 100
17514089 -> 17514191: gap = 102
17573477 -> 17573579: gap = 102
17766599 -> 17766701: gap = 102
17823581 -> 17823683: gap = 102
17826733 -> 17826833: gap = 100
17844361 -> 17844461: gap = 100
17869591 -> 17869693: gap = 102
18042361 -> 18042463: gap = 102
18069371 -> 18069473: gap = 102
18112019 -> 18112121: gap = 102
18118367 -> 18118469: gap = 102
18153547 -> 18153649: gap = 102
18158731 -> 18158831: gap = 100
18341161 -> 18341261: gap = 100
18366133 -> 18366233: gap = 100
18475663 -> 18475763: gap = 100
18494401 -> 18494501: gap = 100
18687169 -> 18687269: gap = 100
18719983 -> 18720083: gap = 100
18751441 -> 18751541: gap = 100
18785467 -> 18785567: gap = 100
19158547 -> 19158649: gap = 102
19161631 -> 19161731: gap = 100
19188649 -> 19188751: gap = 102
19266127 -> 19266229: gap = 102
19282741 -> 19282843: gap = 102
19294423 -> 19294523: gap = 100
19304167 -> 19304267: gap = 100
19415699 -> 19415801: gap = 102
19416511 -> 19416611: gap = 100
19553129 -> 19553231: gap = 102
19643839 -> 19643941: gap = 102
19759177 -> 19759279: gap = 102
19871281 -> 19871381: gap = 100
19896707 -> 19896809: gap = 102
19917067 -> 19917167: gap = 100
19954471 -> 19954573: gap = 102
19956227 -> 19956329: gap = 102
19998047 -> 19998149: gap = 102
20122241 -> 20122343: gap = 102
20256967 -> 20257067: gap = 100
20304313 -> 20304413: gap = 100
20315917 -> 20316017: gap = 100
20583601 -> 20583701: gap = 100
20583709 -> 20583809: gap = 100
20664139 -> 20664239: gap = 100
20737259 -> 20737361: gap = 102
20781931 -> 20782031: gap = 100
20796077 -> 20796179: gap = 102
20851109 -> 20851211: gap = 102
20892757 -> 20892857: gap = 100
21007321 -> 21007423: gap = 102
21024643 -> 21024743: gap = 100
21041879 -> 21041981: gap = 102
21110263 -> 21110363: gap = 100
21146399 -> 21146501: gap = 102
21177139 -> 21177239: gap = 100
21222043 -> 21222143: gap = 100
21242237 -> 21242339: gap = 102
21354451 -> 21354551: gap = 100
21369109 -> 21369209: gap = 100
21425893 -> 21425993: gap = 100
21550939 -> 21551039: gap = 100
21652537 -> 21652639: gap = 102


etc.

The following Java code will list all prime pairs with the given gap:

import java.util.List;
import java.util.ArrayList;

public class Prime100 {
private List<Long> primes = new ArrayList<>();
private long targetGap;

public Prime100(long targetGap) {
this.targetGap = targetGap;
}

boolean isPrime(long n) {
boolean ok = true;
for(long p: primes) {
if(p * p > n) {
break;
}
if(n % p == 0) {
ok = false;
break;
}
}
if(ok) {
long lastPrime = primes.get(primes.size() - 1);
long gap = n - lastPrime;
if(gap == targetGap) {
System.out.println(lastPrime + " -> " + n + ": gap = " + gap);
}

}
return ok;
}

public static void main(String[] args) {
Prime100 primeChecker = new Prime100(102);
long n = 5;
while(true) {
primeChecker.isPrime(n++);
}
}

}


In the following line just replace 102 with the gap you are interested in:

Prime100 primeChecker = new Prime100(102);

• Thank you! Is there any mathematical way? without help of program. Aug 7, 2018 at 10:17
• I don't think so.
– Saša
Aug 7, 2018 at 13:03
• According to the paper First Occurrence Prime Gaps, by Dr. Thomas R. Nicely, the first prime gap of exactly $102$ is from $1,444,309$ to $1,444,411$, discovered bt Glaisher in 1877, Sep 6, 2018 at 8:54
• @DanielWainfleet Nice (and in sync with the output form my script)! I can only imagine how difficult it was in 1877.
– Saša
Sep 6, 2018 at 11:50
• I don't think there can be any solution without programming. I have translated this code to Python, one can check here Mar 29, 2019 at 11:47

See OEIS A000230 for smallest gaps of a given size including some programs. Can we do it without a computer? Sure if you have enough patience and concentration, but I believe historically these people were called computers, so .... But if the question is whether there is a nice mathematical shortcut, no. We need to exhaustively search or use results of other people's exhaustive searches.

Simple programs for finding lots of gaps of a given size. These will be a lot faster than those using trial division.

# Replace '200' with your chosen gap size, and 1e10 with how far you care to look.

perl -E 'use ntheory ":all"; my($$g,$$l)=(200,0); forprimes { say $$l if$$_-$$l==$$g; $$l=$$_; } 1e10'