Consecutive Numbers and Consecutive Primes

Let's take triplets of primes, starting with 2, 3 and 5, and find three consecutive numbers (let's call them \(k, k+1\) and \(k+3\)) that are divisible by these primes respectively. In other words, we need \(k\) to be a multiple of 2, \(k+1\) to be a multiple of 3 and \(k+2\) to be a multiple of 5. A little trial error shows that the value of \(k\) that satisfies is 8 because 2 divides 8, 3 divides 9 and 5 divides 10. This triplet of numbers (8, 9 and 10) is the smallest that is divisible by 2, 3 and 5 respectively. As the primes get larger however, we need an algorithm to find the values of \(k, k+1\) and \(k+2\) that are divisible by consecutive primes \(p, q\) and \(r\). Here is the permalink to an algorithm that will do just that. The results are shown below.

  prime 1   prime 2   prime 3   number 1   number 2   number 3

  2         3         5         8          9          10
  3         5         7         54         55         56
  5         7         11        20         21         22
  7         11        13        791        792        793
  11        13        17        1936       1937       1938
  13        17        19        169        170        171
  17        19        23        4046       4047       4048
  19        23        29        114        115        116
  23        29        31        9453       9454       9455
  29        31        37        31929      31930      31931
  31        37        41        23901      23902      23903
  37        41        43        2664       2665       2666
  41        43        47        44977      44978      44979
  43        47        53        65188      65189      65190
  47        53        59        122482     122483     122484
  53        59        61        134991     134992     134993
  59        61        67        170982     170983     170984
  61        67        71        220027     220028     220029
  67        71        73        101103     101104     101105
  71        73        79        85555      85556      85557
  73        79        83        27886      27887      27888
  79        83        89        296724     296725     296726
  83        89        97        629140     629141     629142
  89        97        101       154326     154327     154328
  97        101       103       546207     546208     546209
  101       103       107       46864      46865      46866
  103       107       109       950587     950588     950589
  107       109       113       1043892    1043893    1043894
  109       113       127       1548890    1548891    1548892
  113       127       131       70738      70739      70740
  127       131       137       702945     702946     702947
  131       137       139       2389964    2389965    2389966
  137       139       149       1513987    1513988    1513989
  139       149       151       416305     416306     416307
  149       151       157       3386174    3386175    3386176
  151       157       163       3220226    3220227    3220228
  157       163       167       1531221    1531222    1531223
  163       167       173       2865051    2865052    2865053
  167       173       179       4309602    4309603    4309604
  173       179       181       3968966    3968967    3968968
  179       181       191       826264     826265     826266
  181       191       193       3557374    3557375    3557376
  191       193       197       2119718    2119719    2119720
  193       197       199       4096811    4096812    4096813
  197       199       211       1823038    1823039    1823040
  199       211       223       8583268    8583269    8583270
  211       223       227       7453997    7453998    7453999
  223       227       229       9175112    9175113    9175114
  227       229       233       9590977    9590978    9590979
  229       233       239       3294852    3294853    3294854

The number associated with my diurnal age today, 27886, appears in the above list and is associated with the primes 73, 79 and 83. Thus we have:$$ \begin{align} 27886 &= 2 \times \textbf{73} \times 191 \\ 27887 &= \textbf{79} \times 353 \\ 27888 &= 2^4 \times 3 \times 7 \times \textbf{83} \end{align} $$Figure 1 shows a plot of the prime 1 against number 1 with some annotations added. Though the trend of number 1's is upward, there is a lot of up and down along the way. The vertical axis of the graph is logarithmic.

Figure 1