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Prime Numbers in a Ten (Posted on 2006-11-06) Difficulty: 3 of 5
In the first decade of numbers (1-10), there are four prime numbers (2,3,5,7). In the second decade (11-20), there are another four (11,13,17,19).

Are there other such decades with four prime numbers?

See The Solution Submitted by Paddy    
Rating: 1.7500 (4 votes)

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Solution re: spoiler -- more | Comment 3 of 5 |
(In reply to spoiler by Robby Goetschalckx)

   101             103             107             109
   191             193             197             199
   821             823             827             829
  1481            1483            1487            1489
  1871            1873            1877            1879
  2081            2083            2087            2089
  3251            3253            3257            3259
  3461            3463            3467            3469
  5651            5653            5657            5659
  9431            9433            9437            9439
 13001           13003           13007           13009
 15641           15643           15647           15649
 15731           15733           15737           15739
 16061           16063           16067           16069
 18041           18043           18047           18049
 18911           18913           18917           18919
 19421           19423           19427           19429
 21011           21013           21017           21019
 22271           22273           22277           22279
 25301           25303           25307           25309
 31721           31723           31727           31729
 34841           34843           34847           34849
 43781           43783           43787           43789
 51341           51343           51347           51349
 55331           55333           55337           55339
 62981           62983           62987           62989
 67211           67213           67217           67219
 69491           69493           69497           69499
 72221           72223           72227           72229
 77261           77263           77267           77269
 79691           79693           79697           79699
 81041           81043           81047           81049
 82721           82723           82727           82729
 88811           88813           88817           88819
 97841           97843           97847           97849
 99131           99133           99137           99139
101111          101113          101117          101119
109841          109843          109847          109849
116531          116533          116537          116539
119291          119293          119297          119299
122201          122203          122207          122209
135461          135463          135467          135469
144161          144163          144167          144169
157271          157273          157277          157279
165701          165703          165707          165709
166841          166843          166847          166849
   10   N1=11:N2=13:N3=17:N4=19
   15   while 1=1
   20    N5=nxtprm(N4)
   30    N1=N2:N2=N3:N3=N4:N4=N5
   40    if N4-N1<10 then print N1,N2,N3,N4:Ct=Ct+1
   45    if Ct>45 then end
   50   wend

This program did not specifically require that the first of the four primes in each group must end in a digit 1, but they do. So you can see that such groups are possible only with the first of them ending in a digit 1, and the others, in turn, must end in 3, 7 and 9. Inclusion of a number ending in 5 is impossible, as such a number would be divisible by 5, and not be prime (outside the first decade). A sequence such as ...3, ...7, ...9, ...1, in that order, would also be impossible, as one of the last three has to be divisible by 3. Similarly for ...7, ...9, ...1, ...3 and ...9, ...1, ...3, ...7.


  Posted by Charlie on 2006-11-06 09:13:22
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