Problems & Puzzles: Puzzles

Puzzle 427. Runs of consecutive numbers such that... (I) Enoch Haga sends the following nice puzzle. Q1) Find a run of 9 or more consecutive integers each having 2 distinct prime factors without repetition -- or prove that it cannot exist (See A064709). Q2) Find a run of 17 or more consecutive integers each having 3 distinct prime factors without repetition -- or prove that such cannot exist (See A080569)

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Contribution came from Frederick Schneider.

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Fred wrote, on Jan 5, 2008:

Q1) I raised the bar to 10^700. It seems quite unlikely that there is an answer.

My approach: Firstly, only one number of the 9 can be a multiple of
six, otherwise you would have abs(2*3^a-2^b*3) = 6 =>
abs(3^(a-1)-2^(b-1))=1 It is well-known that the only solution of this
equation is abs(3^2 - 2^3) = 1

So, you only need to check numbers of the form n=2^x*3^y. I
specifically searched for numbers nearby numbers of the following:

n-2 = 2^f * p1^g
n+2 = 2^j * p2^k
n-3 = 3^l * p3^m
n+3 = 3^p * p4^q

The last number of this type through 10^700 is 169075682574336=2^33 *
3^9. No number through this point as Enoch found, results in 9
consecutive numbers where omega = 2.

Since it is so difficult to find 5 of a potential set of 9, it seems
quite unlikely that a solution exists.


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Q2) I used a sieve to weed out numbers that more 3 or more factors
through a certain prime p (and then went back checked p-smooth numbers
n where omega(n)=3) to find these minimum solutions:

Minimum 17 Solution:

29138958036=2^2 * 3^3 * 269805167
29138958037=7^2 * 23 * 25855331
29138958038=2 * 41911 * 347629
29138958039=3 * 1103 * 8805971
29138958040=2^3 * 5 * 728473951
29138958041=43 * 3319 * 204173
29138958042=2 * 3 * 4856493007
29138958043=13 * 4691 * 477821
29138958044=2^2 * 7 * 1040677073
29138958045=3^2 * 5 * 647532401
29138958046=2 * 11 * 1324498093
29138958047=89 * 443 * 739061
29138958048=2^5 * 3 * 303530813
29138958049=19 * 3533 * 434087
29138958050=2 * 5^2 * 582779161
29138958051=3 * 7 * 1387569431
29138958052=2^2 * 17 * 428514089

Minimum 18 Solution:

146216247221=11 * 19 * 699599269
146216247222=2 * 3 * 24369374537
146216247223=17 * 6679 * 1287761
146216247224=2^3 * 97 * 188422999
146216247225=3 * 5^2 * 1949549963
146216247226=2 * 7 * 10444017659
146216247227=617 * 827 * 286553
146216247228=2^2 * 3^3 * 1353854141
146216247229=13 * 131 * 85858043
146216247230=2 * 5 * 14621624723
146216247231=3 * 367 * 132803131
146216247232=2^6 * 11 * 207693533
146216247233=7 * 409 * 51070991
146216247234=2 * 3 * 24369374539
146216247235=5 * 14561 * 2008327
146216247236=2^2 * 11633 * 3142273
146216247237=3^2 * 12919 * 1257547
146216247238=2 * 4507 * 16221017

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On Oct 5, 2024 Fred wrote again:

The minimum length 19 solution is: 23,671,413,563,491

 

23671413563491 = 83 * 487 * 585621671
23671413563492 = 2^2 * 174749 * 33864877
23671413563493 = 3 * 61673 * 127940447
23671413563494 = 2 * 149 * 79434273703
23671413563495 = 5 * 23^2 * 8949494731

 

23671413563496 = 2^3 * 3 * 986308898479
23671413563497 = 7 * 19 * 177980553109
23671413563498 = 2 * 73 * 162132969613
23671413563499 = 3^2 * 13 * 202319774047
23671413563500 = 2^2 * 5^3 * 47342827127

 

23671413563501 = 11 * 275711 * 7805081
23671413563502 = 2 * 3 * 3945235593917
23671413563503 = 1373 * 3449 * 4998739
23671413563504 = 2^4 * 7 * 211351906817
23671413563505 = 3 * 5 * 1578094237567

 

23671413563506 = 2 * 4517 * 2620258309
23671413563507 = 17 * 43 * 32382234697
23671413563508 = 2^2 * 3^2 * 657539265653
23671413563509 = 29 * 64871 * 12582751

 

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The minimum length 20 solution is:  36,966,736,685,739 . 
 

 

36966736685739 = 3 * 7 * 1760320794559

36966736685740 = 2^2 * 5 * 1848336834287
36966736685741 = 29 * 223 * 5716211023
36966736685742 = 2 * 3 * 6161122780957

36966736685743 = 2591 * 60617 * 235369

 

36966736685744 = 2^4 * 259121 * 8916379
36966736685745 = 3^2 * 5 * 821483037461
36966736685746 = 2 * 7 * 2640481191839
36966736685747 = 11 * 17 * 197683083881

36966736685748 = 2^2 * 3 * 3080561390479

 

36966736685749 = 13 * 257 * 11064572489
36966736685750 = 2 * 5^3 * 147866946743
36966736685751 = 3 * 79 * 155977791923
36966736685752 = 2^3 * 2333 * 1980643843
36966736685753 = 7 * 16421 * 321598099
 

 

36966736685754 = 2 * 3^2 * 2053707593653

36966736685755 = 5 * 1499 * 4932186349
36966736685756 = 2^2 * 63397 * 145774787
36966736685757 = 3 * 19 * 648539240101

36966736685758 = 2 * 11 * 1680306212989

 

There are no longer sequences under 10^15.

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