Take the first $n$ primes $p\_1,...,p\_n$ and the primorial $P\_n$ .Denote by $p\_i$ every prime bigger than $p\_n$ and smaller than $P\_n$.


1) Is that true that there always be a number in any interval of consecutive integers of length $P\_n$ not divided by any $p\_i$? (It's the same as taking a residue class $r\_i\bmod p\_i$ for every $p\_i$ in every possible way and wondering if you can cover all the numbers in the interval $[0,P\_n-1]$.)


ADDED:


2) Even if we do not know if we can cover this interval, can we have any good upper bound on the number of ways?



residue classes of primes, covering intervals and bounds on the different ways
I did some computer programming to check plausibility. In future I request that you do this step yourself.


For $p\_n = 3$ and $P\_n = 6,$ the only prime in between is 5, and any interval of length 6 contains an integer not congruent to any prescribed value mod 5. 


In C++ I was able to check up to 10,000,000. For definiteness I took the residue classes to all be 0, that is I checked multiples of the primes between $p\_n$ and $P\_n.$ For the $p\_n$ I checked, I was able to find only relatively short intervals of consecutive numbers, each of which is divisible by at least one prime between $p\_n$ and $P\_n.$ That is, these intervals have lengths much shorter than $P\_n$ itself. Thus in any interval of length $P\_n,$ it should be quite easy to find numbers that are not divisible by any of those primes. Indeed, the probability of picking a success at random appears to increase with $p\_n.$


For example, for $p\_n = 5, P\_n = 30,$ I tried to find long intervals where each number had at least one divisor in the set 7, 11, 13, 17, 19, 23, 29. 



```
 691558 = 2 * 7 * 47 * 1051
 691559 = 11 * 62869
 691560 = 2^3 * 3^2 * 5 * 17 * 113
 691561 = 13 * 53197
 691562 = 2 * 19 * 18199
 691563 = 3 * 29 * 7949
 691564 = 2^2 * 23 * 7517
 691565 = 5 * 7 * 19759

```

The bound of 10,000,000 is not on primes, it is on the output, such as 691565 < 10,000,000.



```
  p_n = 5       P_n = 30    0.592302   2.96151
 length = 8
  691558  691559  691560  691561  691562  691563  691564  691565


  p_n = 7    P_n = 210  0.454539   3.18177 
  length = 20
  635088  635089  635090  635091  635092  635093  635094  635095  635096  635097
  635098  635099  635100  635101  635102  635103  635104  635105  635106  635107

    p_n = 11     P_n = 2310   0.348014   3.82815
length = 43
    2113    2114    2115    2116    2117    2118    2119    2120    2121    2122
    2123    2124    2125    2126    2127    2128    2129    2130    2131    2132
    2133    2134    2135    2136    2137    2138    2139    2140    2141    2142
    2143    2144    2145    2146    2147    2148    2149    2150    2151    2152
    2153    2154    2155

    p_n = 13     P_n = 30030   0.283807   3.68949
length = 207
   29745   29746   29747   29748   29749   29750   29751   29752   29753   29754
   29755   29756   29757   29758   29759   29760   29761   29762   29763   29764
   29765   29766   29767   29768   29769   29770   29771   29772   29773   29774
   29775   29776   29777   29778   29779   29780   29781   29782   29783   29784
   29785   29786   29787   29788   29789   29790   29791   29792   29793   29794
   29795   29796   29797   29798   29799   29800   29801   29802   29803   29804
   29805   29806   29807   29808   29809   29810   29811   29812   29813   29814
   29815   29816   29817   29818   29819   29820   29821   29822   29823   29824
   29825   29826   29827   29828   29829   29830   29831   29832   29833   29834
   29835   29836   29837   29838   29839   29840   29841   29842   29843   29844
   29845   29846   29847   29848   29849   29850   29851   29852   29853   29854
   29855   29856   29857   29858   29859   29860   29861   29862   29863   29864
   29865   29866   29867   29868   29869   29870   29871   29872   29873   29874
   29875   29876   29877   29878   29879   29880   29881   29882   29883   29884
   29885   29886   29887   29888   29889   29890   29891   29892   29893   29894
   29895   29896   29897   29898   29899   29900   29901   29902   29903   29904
   29905   29906   29907   29908   29909   29910   29911   29912   29913   29914
   29915   29916   29917   29918   29919   29920   29921   29922   29923   29924
   29925   29926   29927   29928   29929   29930   29931   29932   29933   29934
   29935   29936   29937   29938   29939   29940   29941   29942   29943   29944
   29945   29946   29947   29948   29949   29950   29951

     p_n = 17     P_n = 510510   0.236611   4.0224
   length = 1 + 435005 -  433756 = 1250
   433756  433757  433758  433759  433760  433761  433762  433763  433764  433765
....
...
   434996  434997  434998  434999  435000  435001  435002  435003  435004  435005.

```

