Special whole numbers...

[mfgoode]

Wpolly: you are slightly off the mark. There is at least one and maybe more 13 digit prime(s) less than 1,000,000,000,063.
hint: you wont have to go far to find it!

Your number 196883: =196,560 + 323
196,560 is the number of spheres touching any one sphere in a 24-dimensional Leech lattice
323 = 17*19
However let's have your answer wpolly so the number is up again

196883.
Mally

[mfgoode]

Originally Posted by Kees

Product of three primes in arithmetic progression (47,59,71)

New number: 30042907

30042907 = 109 *275623 both primes.

30042907 Ive put the number up again. Lets hear from you Kees.
Mally

[Kees]

smallest 13 digit prime is 1,000,000,000,039.

For 30042907 as a hint, think about corollaries to fermats last theorem.

[mfgoode]

You are right on 1,000,000,000,039.It is indeed a prime. But is there one further up below 1,000,000,000,063 ?
Mally

[Kees]

Well, 1,000,000,000,061 is prime two, so we have twins

[Kees]

And if we look a little bit further we find another one, giving
the primes

1,000,000,000,061
1,000,000,000,091
1,000,000,000,121

[fetofs]

Originally Posted by Kees

smallest 13 digit prime is 1,000,000,000,039.

For 30042907 as a hint, think about corollaries to fermats last theorem.

1+2^3 = 3^2
2^5+7^2 = 3^4
7^3+13^2 = 2^9
2^7+17^3 = 71^2
3^5+11^4 = 122^2
17^7+76271^3 = 21063928^2
1414^3+2213459^2 = 65^7
9262^3+15312283^2 = 113^7
43^8+96222^3 = 30042907^2
33^8+1549034^2 = 15613^3

As for the number, I'll leave wpollly's one.

EDIT: But, as we expect her answer (which probably is the one Kees gave):

278914005382139703576000

[mfgoode]

Originally Posted by Kees

Well, 1,000,000,000,061 is prime two, so we have twins

The reason why I gave 1........63 is because these are easily remembered twin primes in 13 digit primes and 13 itself is a prime.
Well we now have fetofs 278914005382139703576000
Mally

[Kees]

fetofs was spot on for my number.
His number factors as

(2^6)*(3^5)*(5^3)*(7^2)*11*13*17*19*23*29*31*37*41*43*47

which uses al the prime numbers under 50. But this probably is not the answer
due to the extra factors of 2, 3, 5 and 7.
But then again, it is still early morning for me

[mfgoode]

Originally Posted by Kees

fetofs was spot on for my number.
His number factors as

(2^6)*(3^5)*(5^3)*(7^2)*11*13*17*19*23*29*31*37*41*43*47

which uses al the prime numbers under 50. But this probably is not the answer
due to the extra factors of 2, 3, 5 and 7.
But then again, it is still early morning for me

So what of it?! To me it is perfectly acceptable Kees. Its a remarkable piece of factorisation. What URL do you use ?
Mally

[Kees]

Not entirely sure that I understood your question properly, but for factorisation
PARI does quite a nice job upto rather large numbers. But I really think that fetops has something else in mind, therefore I have not posted another number yet.