B"H
The definition of a sphenic number is that it is the product of three distinct prime numbers, none of them repeating themselves. 30 is the first. 42, 66, 70, 78...
30 is 2*3*5. 42 is 2*3*7...you get the picture. The cool thing about base 10 is that 10 is the product of two primes. Thus, if we take any prime number and multiply it by 10, we get a sphenic number, excluding 20 and 50 of course.
30 is 3*2*5. 70 is 7*2*5. 110 is 11*2*5. Base 6 would work also. Base 15, Base 21...
It is coincidental, but you can take all prime numbers, excluding 20 and 50, slap a zero after them, and you have a sphenic number. I'm not sure why this is so cool, but it is.
Yes, as you can tell, I am tired. I really had better go to bed. No, this is not Aspie-ness that would cause me to take note of this. It is probably plain craziness...
Good night!
It is coincidental, but you can take all prime numbers, excluding 20 and 50, slap a zero after them, and you have a sphenic number.
Not really coincidental, since putting a zero on the end is equivalent to multiplying by 2 times 5.
Here's an interesting (for me) fact about prime numbers: all prime numbers greater than three give a remainder of either 1 or 5 when divided by 6. I wish I could say I figured this out for myself, but in fact I read it in God's Secret Formula by Peter Plichta.
@Aeolienne it's even better to say 1 or -1 than 1 or 5....
Or I could say that the absolute value of all primes greater than 3 is equal to 1 modulo 6.
Or I could say that the absolute value of all primes greater than 3 is equal to 1 modulo 6.
That one doesn't really work - modulo 6, the answer would be 5 rather than abs(-1).
Ditto with remainders - you can't have a remainder of -1.
B"H
Aeolienne,
By "coincidental" I merely meant that base ten just happens to be a base that is a multiple of two distinct primes. Base 6 would have the same rule, just add a zero and you get a Sphenic. Base 8 would not...
Thanks for posting.
PS. I am not formally educated in number theory, so I will ask if that is a proven theorem or an unproven conjecture like Goldbach's?
It sounds like something that derives from the fact that 6 is a product of 2 and 3, the first two primes. Oddly, I have considered the "prime pairs," and realize that those I know are indeed one before and one after multiples of 6. Someone let me know where to find the proof of this one? Wolfram?
I admit that I am not formally educated in number theory not out of humility, but because I'd rather you know that than think I was asleep!
B"H
OK, here goes one proof:
http://www.mathpages.com/home/kmath073/kmath073.htm
You mathematics pro's will understand it better than I. It is based on Euler's totient function. Of course, there are laws about primes going back to the Greeks that I can barely understand how they knew to arrive at!
All the best,
Ditto with remainders - you can't have a remainder of -1.
Depends on how you define a remainder. While it's a convention they mut be positive (because then you get the integer part before the remainder), negative remainders make sense too, I mean you can calculate with them just like with the positive one and still get plausible results.
B"H
Well, you can argue that a remainder of 1 and 5 mean exactly the same thing as 1 and -1 when considering multiples of 6. However, I have generally seen positive remainders when the whole "modulus" issue comes up. However, the proof I linked to also showed a -1. So, now you all have me curious!
Here is a possible tie-breaker. One *USED* to be considered a prime, and does have some qualities of a prime. So, one is one more than zero, which is a multiple of 6, along with every other real number. Yet, -1 is not considered a prime, nor is any negative number, unless we really want to open up a can of worms. Thus, I would probably keep my notation fixed with positive numbers in this case. I am a non-mathematician speaking, of course.
Honestly, I did not fall asleep. I never had the course. I couldn't find the classroom because it was "7 mod(6)+1" and that number would not have made sense until I took the course. Thus, I had a paradoxical time running around the campus.(-;
B"H
OK, I feel dumb. The proof is very obvious. 6 is a multiple of 2 and 3. Every multiple of six is also a multiple of 2 and 3, both primes. Thus, two and four more than any multiple of six would be multiples of two. 3 more than any multiple of six would be multiples of 3. This leaves only one more and five more that could *potentially* be primes, just as Eratosthenes' Sieve demonstrated primes by knocking off composites.
I thought of this while running. I did not need to scour the Net for proofs. Oh well...
All the best,