Re: [PrimeNumbers] Fun with (eeek!) composites

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• ... 2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14 See http://www.primepuzzles.net/puzzles/puzz_181.htm 5,516,280 + k is the smallest
Message 1 of 3 , Apr 4 4:13 PM
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Mark Underwood wrote:

> Fun with successive numbers:
>
>
> 18,164,161 is a prime times 1
...
> 18,164,167 is a prime times 7.

2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14
See http://www.primepuzzles.net/puzzles/puzz_181.htm
5,516,280 + k is the smallest solution for k = 1..7

> 34,415,168 is divisible by 2
> 34,415,169 is divisible by 3
> 34,415,170 is divisible by 5
> 34,415,171 is divisible by 7
> 34,415,172 is divisible by 11
> 34,415,173 is divisible by 13
> 34,415,174 is divisible by 17
> 34,415,175 is divisible by 19
> 34,415,176 is divisible by 23.

This is what CRT (Chinese Remainder Theorem) is for. A bigint program can easily
reach primes in the millions but I will spare you solutions with millions of
digits.

> 12,252,242 is divisible by 2
...
> 12,252,258 is divisible by 18.

Because lcm(2,...,18) = 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 = 12,252,240
(lcm = least common multiple)

My contribution to fun with successive numbers is the smallest case of 10
numbers where the nth has n prime factors:

3931520917431241 = 3931520917431241
3931520917431242 = 2 * 1965760458715621
3931520917431243 = 3 * 221477 * 5917124453
3931520917431244 = 2 * 2 * 23 * 42733923015557
3931520917431245 = 5 * 17 * 199 * 5557 * 41826179
3931520917431246 = 2 * 3 * 29 * 83 * 261799 * 1039837
3931520917431247 = 7 * 7 * 7 * 19 * 43 * 557 * 25187741
3931520917431248 = 2 * 2 * 2 * 2 * 31 * 59 * 167 * 804471071
3931520917431249 = 3 * 3 * 3 * 3 * 3 * 41 * 53 * 1721 * 4326271
3931520917431250 = 2 * 5 * 5 * 5 * 5 * 5 * 11 * 13 * 13 * 338377271

The nth number is divisible by n above, but not in the 2nd smallest case
starting at 5,818,684,827,160,441.

See http://www.research.att.com/projects/OEIS?Anum=A072875

--
Jens Kruse Andersen
• Hi Jens The (N+k)/k prime puzzle. If I ve read it before I don t remember it! Perhaps it was posted on this site a year ago or so when I was visiting, and it
Message 2 of 3 , Apr 4 6:30 PM
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Hi Jens

The (N+k)/k prime puzzle. If I've read it before I don't remember it!
Perhaps it was posted on this site a year ago or so when I was
visiting, and it lodged in my subconscious :)

I couldn't believe I sent the 12,252,240 + n is divisible by n thing.
I realized a minute after I sent it what I had done, but it was too
late!

The Nth number has N prime factors puzzle is brilliant.
That the Nth number is also divisible by N is not too improbable, on
inspection. The 2,3,6 8th and 9th numbers *must* be divisible by N.
Getting both the 5th and 10th numbers divisible by 5 and 10 is about
a four in five chance, assuming there is not some unknown principle
which might make it more certain or even guaranteed. Getting the
seventh number divisible by 7 is about fifty fifty, again assuming
there is not some unknown princple stacking the deck further.

Mark

--- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
<jens.k.a@g...> wrote:
> Mark Underwood wrote:
>
> > Fun with successive numbers:
> >
> >
> > 18,164,161 is a prime times 1
> ...
> > 18,164,167 is a prime times 7.
>
> 2,918,756,139,031,688,155,200 + k is a prime times k, for k = 1..14
> See http://www.primepuzzles.net/puzzles/puzz_181.htm
> 5,516,280 + k is the smallest solution for k = 1..7
>
> > 34,415,168 is divisible by 2
> > 34,415,169 is divisible by 3
> > 34,415,170 is divisible by 5
> > 34,415,171 is divisible by 7
> > 34,415,172 is divisible by 11
> > 34,415,173 is divisible by 13
> > 34,415,174 is divisible by 17
> > 34,415,175 is divisible by 19
> > 34,415,176 is divisible by 23.
>
> This is what CRT (Chinese Remainder Theorem) is for. A bigint
program can easily
> reach primes in the millions but I will spare you solutions with
millions of
> digits.
>
> > 12,252,242 is divisible by 2
> ...
> > 12,252,258 is divisible by 18.
>
> Because lcm(2,...,18) = 2^4 * 3^2 * 5 * 7 * 11 * 13 * 17 =
12,252,240
> (lcm = least common multiple)
>
>
> My contribution to fun with successive numbers is the smallest case
of 10
> numbers where the nth has n prime factors:
>
> 3931520917431241 = 3931520917431241
> 3931520917431242 = 2 * 1965760458715621
> 3931520917431243 = 3 * 221477 * 5917124453
> 3931520917431244 = 2 * 2 * 23 * 42733923015557
> 3931520917431245 = 5 * 17 * 199 * 5557 * 41826179
> 3931520917431246 = 2 * 3 * 29 * 83 * 261799 * 1039837
> 3931520917431247 = 7 * 7 * 7 * 19 * 43 * 557 * 25187741
> 3931520917431248 = 2 * 2 * 2 * 2 * 31 * 59 * 167 * 804471071
> 3931520917431249 = 3 * 3 * 3 * 3 * 3 * 41 * 53 * 1721 * 4326271
> 3931520917431250 = 2 * 5 * 5 * 5 * 5 * 5 * 11 * 13 * 13 * 338377271
>
> The nth number is divisible by n above, but not in the 2nd smallest
case
> starting at 5,818,684,827,160,441.
>
> See http://www.research.att.com/projects/OEIS?Anum=A072875
>
> --
> Jens Kruse Andersen
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