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)

Contribution cam from Frederick Schneider.

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Fred wrote:

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.

==============================================

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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