Posted: Apr 10, 2010 1:56 am
A reminder that I will only be reading what is said in the interim after I finish all of these posts.
First a note of correction: My most recent submission to the Prime Curios site, 126704222713 being the smallest prime p such that the sums of the nth powers of all the primes up to and including p for n=1 to 6 are prime, has been accepted. Whatever was preventing the acceptance of others might be gone or have been misinterpreted. Now I'm guessing the prof wants proof of more technical facility from me rather than freaky stuff. Who knows?
Now, I'll start with the second largest of my accepted submissions. It's the analogue of 82818079...321 in base 7. The base-ten value of the starting number is 373, i.e. the prime is 1042104110401036...3217. I hadn't even previously considered this as in any way having to do with base-ten coincidence, but it does tie in a little bit since 73=343. It also offsets nicely from the one coincidence on 151 (in the quote of the first of this series toward the end), though they are hardly related except as minor parts of something more general about palindromes in base ten.
Now, I said 71 reminded me, so I'll look at one of the two submissions on 17. The one that reminded me is that the number of ways that 71 converts to a prime in going from smaller base to larger with larger base no greater than ten is 17, and this makes 71 a record-holder up to a very large number (I've repeatedly asked that my parenthetical on that curio be removed since almost immediately after submitting it, in case you read it, because it's bound to be very wrong). The other curio is a whole complex of stuff, really, that I will save till after I've covered everything else.
999779999159200499899 is the final 21-digits (with 12 nines) of 9999, and is prime. Along with this goes the fact that the first occurrence of seven consecutive digits six of which are the same, excluding terminal 0s from consideration, in a listing of the numbers nn is the string 5555575 at the very middle of 9696.
The curio on 4500 is a little interesting but need not be repeated. The more interesting thing to me about it is that I am (and was when I discovered/submitted it) 45 years old.
The curio on 5 is not particularly anything relating to base ten except that I found it after seeing a curio on 100. The fact that nobody before me apparently submitted it is what is surprising there. The curio is that both (prime(5))!/5!+1 and (prime(5!))!/(5!)!+1 are prime, prime(n) being the nth prime.
The curios on 73 and 173 are scarcely base-ten coincidences separately. They make a tiny impression, but when I get to the next post [I decided to put off the 365 stuff and focus on this 'complex' in the next post], if you have read them, you will see what that impression is. Not all that important. The curio on 1000 is a part of the complex entirely, so I'm saving that; 2437 doesn't seem to have anything directly to do with coincidences, but is merely research submitted and accepted; 17769643 is good nice research, but explaining how (at all) it's a coincidence to me is something I'm apt to mention in passing in the last of this series of posts; the 140-digit one is a part of the complex of results; and 0 is purely a mathematics statement about definitions that was accepted. So discussion of the complex of results is what I'll be doing after a brief relating of a few of my other discoveries and things I've come across (generally the former) .
On palindromes again briefly, the first prime over 1000, 1009, is the smallest number which is a 3-digit palindrome in five different bases, and the smallest number k for which a prime of the form k*2n-1 is currently not known, 2293, is the smallest palindrome of length 5 in two different bases. 1001 appeals to me as the product of three consecutive primes, but whether you make of it a coincidence of any kind is another matter, and it's no discovery. And the fact that 41041, 410041, and 101101 are all Carmichael numbers is something I can't even begin to appreciate, but I mention it since I have it written down on this short list in front of me.
Just a couple more, and then I'll go back to the complex of results. A discovery of mine that was not accepted at the Curios site was that 231661 reads as a prime beginning with base 10 all the way through base 22, excepting base 20. It's the smallest number/string that reads as a prime through base 16, and the smallest that reads as a prime through base 20 is over a thousand times larger.
I'm currently researching beyond a nice coincidence I really have to describe clearly now. If you take four distinct numbers, then there are 12 ways of concatenating pairs. In researching a very specific question, I got a rather remarkable result. The question is what are the record-setting collections of four consecutive primes in terms of the number of ways out of the twelve ways of concatenating pairs that are prime. Well, there is bound to eventually--I think--show on my computer screen a collection that generates ten, but the one generating nine is cool. This is what I'm submitting to Prime Curios right now: "5608951: The first collection of four successive primes giving as many as 9 out of 12 primes by concatenation of pairs is 100 times this prime plus 3 times the 7th-10th primes. The remaining three concatenations are all semiprimes."
Since I will need to send an e-mail along with it, the next post will be about the complex I keep referring to. I should be done with all of this in about 8 hours (I think).
Edit: I entirely forgot an interesting/strange one and something neat on 42 in conjunction with base 10; and I'm sure I have others not having to do with the topics of the concluding posts, but I'm just going to add these.
A. 367434 is the first base in which 21, 321, 4321, 54321, and 654321 prime, and 7654321 is also. This base is II0I27, the 7-digit left-truncatable prime with the smallest digit sum for it is 5132491, this read as base 10 is 3209736, and 3209710=1(100)1136 (By the way, 1001 was found by me--on someone else's list--to be the 136th product of three distinct primes shortly after this whole thing, and my father's d.o.b. is 3 January 1942). The first base for up to 54321 being prime is 4578=AI456, among other things.
B. The smallest prime which splits digitally in some base into three primes so that the other five arrangements are also prime is 4201 (in bases 3 and 9 both) and the smallest in specifically base 84=2*42 is 41999. To be clear about what is meant, 11437 is the number for base 10 as 11743, 43117, 43711, 71143, and 74311 are all prime. This last example is one of the ones rejected. (Oh, well!)
Edit II: I just looked at the details of something. Worthy of mention is the new result by Eric Weisstein: 377653776437763...321 is the second (highly probably) prime in the sequence with 828180...321, and it has 177719 digits.
Edit III: For the curious who are willing to do some work on what-all I can think of, one more thing came up. Look at the 16th-18th members of http://research.att.com/~njas/sequences/A171134 and then look at the similar stuff in the sequence of factorials (noticed by many, I'm sure).
Edit IV: Here's something else--two similar problems:
1) For what numbers a and bases b is (b-a)a a palindrome with digit sum b?
2) For what numbers a and b is (a+b)a-a(b-a)=b?
First a note of correction: My most recent submission to the Prime Curios site, 126704222713 being the smallest prime p such that the sums of the nth powers of all the primes up to and including p for n=1 to 6 are prime, has been accepted. Whatever was preventing the acceptance of others might be gone or have been misinterpreted. Now I'm guessing the prof wants proof of more technical facility from me rather than freaky stuff. Who knows?
Now, I'll start with the second largest of my accepted submissions. It's the analogue of 82818079...321 in base 7. The base-ten value of the starting number is 373, i.e. the prime is 1042104110401036...3217. I hadn't even previously considered this as in any way having to do with base-ten coincidence, but it does tie in a little bit since 73=343. It also offsets nicely from the one coincidence on 151 (in the quote of the first of this series toward the end), though they are hardly related except as minor parts of something more general about palindromes in base ten.
Now, I said 71 reminded me, so I'll look at one of the two submissions on 17. The one that reminded me is that the number of ways that 71 converts to a prime in going from smaller base to larger with larger base no greater than ten is 17, and this makes 71 a record-holder up to a very large number (I've repeatedly asked that my parenthetical on that curio be removed since almost immediately after submitting it, in case you read it, because it's bound to be very wrong). The other curio is a whole complex of stuff, really, that I will save till after I've covered everything else.
999779999159200499899 is the final 21-digits (with 12 nines) of 9999, and is prime. Along with this goes the fact that the first occurrence of seven consecutive digits six of which are the same, excluding terminal 0s from consideration, in a listing of the numbers nn is the string 5555575 at the very middle of 9696.
The curio on 4500 is a little interesting but need not be repeated. The more interesting thing to me about it is that I am (and was when I discovered/submitted it) 45 years old.
The curio on 5 is not particularly anything relating to base ten except that I found it after seeing a curio on 100. The fact that nobody before me apparently submitted it is what is surprising there. The curio is that both (prime(5))!/5!+1 and (prime(5!))!/(5!)!+1 are prime, prime(n) being the nth prime.
The curios on 73 and 173 are scarcely base-ten coincidences separately. They make a tiny impression, but when I get to the next post [I decided to put off the 365 stuff and focus on this 'complex' in the next post], if you have read them, you will see what that impression is. Not all that important. The curio on 1000 is a part of the complex entirely, so I'm saving that; 2437 doesn't seem to have anything directly to do with coincidences, but is merely research submitted and accepted; 17769643 is good nice research, but explaining how (at all) it's a coincidence to me is something I'm apt to mention in passing in the last of this series of posts; the 140-digit one is a part of the complex of results; and 0 is purely a mathematics statement about definitions that was accepted. So discussion of the complex of results is what I'll be doing after a brief relating of a few of my other discoveries and things I've come across (generally the former) .
On palindromes again briefly, the first prime over 1000, 1009, is the smallest number which is a 3-digit palindrome in five different bases, and the smallest number k for which a prime of the form k*2n-1 is currently not known, 2293, is the smallest palindrome of length 5 in two different bases. 1001 appeals to me as the product of three consecutive primes, but whether you make of it a coincidence of any kind is another matter, and it's no discovery. And the fact that 41041, 410041, and 101101 are all Carmichael numbers is something I can't even begin to appreciate, but I mention it since I have it written down on this short list in front of me.
Just a couple more, and then I'll go back to the complex of results. A discovery of mine that was not accepted at the Curios site was that 231661 reads as a prime beginning with base 10 all the way through base 22, excepting base 20. It's the smallest number/string that reads as a prime through base 16, and the smallest that reads as a prime through base 20 is over a thousand times larger.
I'm currently researching beyond a nice coincidence I really have to describe clearly now. If you take four distinct numbers, then there are 12 ways of concatenating pairs. In researching a very specific question, I got a rather remarkable result. The question is what are the record-setting collections of four consecutive primes in terms of the number of ways out of the twelve ways of concatenating pairs that are prime. Well, there is bound to eventually--I think--show on my computer screen a collection that generates ten, but the one generating nine is cool. This is what I'm submitting to Prime Curios right now: "5608951: The first collection of four successive primes giving as many as 9 out of 12 primes by concatenation of pairs is 100 times this prime plus 3 times the 7th-10th primes. The remaining three concatenations are all semiprimes."
Since I will need to send an e-mail along with it, the next post will be about the complex I keep referring to. I should be done with all of this in about 8 hours (I think).
Edit: I entirely forgot an interesting/strange one and something neat on 42 in conjunction with base 10; and I'm sure I have others not having to do with the topics of the concluding posts, but I'm just going to add these.
A. 367434 is the first base in which 21, 321, 4321, 54321, and 654321 prime, and 7654321 is also. This base is II0I27, the 7-digit left-truncatable prime with the smallest digit sum for it is 5132491, this read as base 10 is 3209736, and 3209710=1(100)1136 (By the way, 1001 was found by me--on someone else's list--to be the 136th product of three distinct primes shortly after this whole thing, and my father's d.o.b. is 3 January 1942). The first base for up to 54321 being prime is 4578=AI456, among other things.
B. The smallest prime which splits digitally in some base into three primes so that the other five arrangements are also prime is 4201 (in bases 3 and 9 both) and the smallest in specifically base 84=2*42 is 41999. To be clear about what is meant, 11437 is the number for base 10 as 11743, 43117, 43711, 71143, and 74311 are all prime. This last example is one of the ones rejected. (Oh, well!)
Edit II: I just looked at the details of something. Worthy of mention is the new result by Eric Weisstein: 377653776437763...321 is the second (highly probably) prime in the sequence with 828180...321, and it has 177719 digits.
Edit III: For the curious who are willing to do some work on what-all I can think of, one more thing came up. Look at the 16th-18th members of http://research.att.com/~njas/sequences/A171134 and then look at the similar stuff in the sequence of factorials (noticed by many, I'm sure).
Edit IV: Here's something else--two similar problems:
1) For what numbers a and bases b is (b-a)a a palindrome with digit sum b?
2) For what numbers a and b is (a+b)a-a(b-a)=b?