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 Odd primes never die (Posted on 2010-05-20)
I 've found an interesting table of numbers in an old issue of JMR, dedicated to astounding trivia regarding primes.
Erasing all the digits in the table's footnotes I got a challenging, albeit solvable puzzle:
The XX consecutive primes from X to XX sum up to the prime number XXX.
Also when arranged in groups of three, each group sums up to a prime number.
Furthermore, those partial sums with their digits reversed, also sum up to the same sum as before the reversal!

Try to reconstruct the trivia : both the table and the text.

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 Is this it? | Comment 4 of 11 |

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10   repeat
20     P1=nxtprm(P1):P2=P1:Tot=P1:Ct=1
30     for I=1 to 8:P2=nxtprm(P2):Tot=Tot+P2:Ct=Ct+1:next
40     repeat
50         P2=nxtprm(P2):Ct=Ct+1
60         Tot=Tot+P2
70         if prmdiv(Tot)=Tot and Tot>99 and Tot<1000 and P2<100 and P1<9 the
n
80            :gosub *CheckIt
90     until P2>99
100   until P1>9
110   end
120   *CheckIt
130   ChkTot=0
140   Px=P1:ChkTot=P1:print Px;
150   repeat
160    Px=nxtprm(Px):print Px;
170    if Px<10 then ChkTot=ChkTot+Px
180    if Px>10 then ChkTot=ChkTot+10*(Px@10)+Px\10
190   until Px=P2:print
200   if nxtprm(ChkTot)=ChkTot then print P1;P2,Tot,ChkTot
205                                 print Ct,P1;P2,Tot,ChkTot:print
210   return
OK

The UBASIC program above finds sequences that start with a single digit prime and extend at least 10 primes to a two-digit prime, and add up to a prime.

I've marked off part of its output based on criteria shown below:

`run 2  3  5  7  11  13  17  19  23  29  31  37 12      2  37   197     431`
` 2  3  5  7  11  13  17  19  23  29  31  37  41  43 14      2  43   281     479`
` 3  5  7  11  13  17  19  23  29  31  37  41  43  47  53 15      3  53   379     586`
` 3  5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61 17      3  61   499     697`
` 5  7  11  13  17  19  23  29  31  37  41 11      5  41   233     440`
` 5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67 17      5  67   563     770`
` 7  11  13  17  19  23  29  31  37  41  43 11      7  43   271     469`
`----------------------------------------------------------- 7  11  13  17  19  23  29  31  37  41  43  47  53  59  61 15      7  61   491     689-----------------------------------------------------------`
` 7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  8389 21      7  89   953     1052`

Only multiples of 3 in the count of primes were further tested via spreadsheet, which included the 15-member series marked off above.

`       2      10       3              15                    |371|     13       5                      23            |131|     95       7      31                            | 79|     79      11              41                    | 95|    131      13                      49            | 13|    371      17      59                            *689*    112      19              71                             152      23                      83                     953      29      97      31             109                      38      37                     121              94      41     131                              32      43             143                     164      47                     159      53     173      59             187      61                     199      67     211      71             223      73                     235      79     251      83      89`

The columns on the spreadsheet show the primes from 2 to 89 and the triplet sums starting at any of the three possible starting points.

As the 15-member sequence does indeed have a multiple of three members, the series of triple totals that starts at 7, starts the totals with 31 (=7+11+13) and includes 59, 97, 131 and 173.  The basic program had already calculated the total of the reversed digits primes themselves as 689. Off to the side, on the spreadsheet, the five triplet totals, with digits reversed, also add to 689. I think this is what is sought, but then again what it matches is not the sum of the non-reversed groups of three but rather the reversed-digit individual primes of the 15-prime series.

 Posted by Charlie on 2010-05-20 22:44:12

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