I have a particular penchant for an unusual class of numbers called "exclusionary squares." Can you tell me what is so special about the number
It turns out that this is the largest integer with distinct digits whose square is made up of digits not included in itself: 639,1722 = 408,540,845,584
Can you find the only other six-digit example? Can you find any exclusionary cubes? (I learned about exculsionary squares from my colleague Andy Edwards. So far, I am unaware of any large exclusionary cubes, and we may wish to generalize this to exclusionary numbers of the Nth order. I will report your "world records" in a forthcoming book.)
Ilan Mayer, Toronto, Canada (ilan at cedara dot com) Other 6 digit example: 203879^2 = 41566646641 Exclusionary cubes (search up to 1000000): 2^3 = 8 3^3 = 27 7^3 = 343 8^3 = 512 27^3 = 19683 43^3 = 79507 47^3 = 103823 48^3 = 110592 52^3 = 140608 53^3 = 148877 63^3 = 250047 68^3 = 314432 92^3 = 778688 157^3 = 3869893 172^3 = 5088448 187^3 = 6539203 192^3 = 7077888 263^3 = 18191447 378^3 = 54010152 408^3 = 67917312 423^3 = 75686967 458^3 = 96071912 468^3 = 102503232 478^3 = 109215352 487^3 = 115501303 527^3 = 146363183 587^3 = 202262003 608^3 = 224755712 648^3 = 272097792 692^3 = 331373888 823^3 = 557441767 843^3 = 599077107 918^3 = 773620632 1457^3 = 3092990993 1587^3 = 3996969003 1592^3 = 4034866688 4657^3 = 100999381393 4732^3 = 105958111168 5692^3 = 184414333888 6058^3 = 222324747112 6378^3 = 259449922152 7658^3 = 449103134312 A Related Problem Jonathan Dushoff: >>> If you drop the requirement that digits in the original be >>> distinct you can find exclusionary squares of any length. Mark Brader: >> Specifically? > Ted Schuerzinger (and Michael Crowder in email): > Any number of 3's. Right. For example, 3333333^2 = 11111108888889. But there are no other such series, because * 1^2 ends in 1 * 222^2 ends in 284 * 444444^2 ends in 469136 * 5^2 ends in 5 * 6^2 ends in 6 * 777^2 ends in 603729 * 888888^2 ends in 876544 * each of 99^2, 999^2, 9999^2, etc. starts with 9. -- Mark Brader, Toronto, firstname.lastname@example.org > Right. For example, 3333333^2 = 11111108888889. But there are no > other such series, because ... True, but only for a narrow definition of "such series". For example 66...7 works, as does 33...7. Are there any with longer repeating components, or with more than two distinct digits? Jonathan Dushoff Jonathan Dushoff writes: > True, but only for a narrow definition of "such series". For example > 66...7 works, as does 33...7. It's narrow to say that these don't fall under the rubric of "integers *without* distinct digits"?? -- Mark Brader Martin Round: 69^2 = 4761 69^3 = 328509 The square and the cube are exclusionary. I supose it's unique in that all ten digits are used. "William Rex Marshall"
: > 69^2 = 4761 ^ ^ > 69^3 = 328509 ^ ^ These are not exclusionary. "Martin Round" Sorry. I didn't make myself clear.
The two numbers 4761 and 328509 use each of the ten digits 0,1,2,3,4,5,6,7,8,9 once and only once.
4761 is 69 squared and 328509 is 69 cubed.
I feel pretty confident that the number 69 is unique in that it generates a square and a cube that are exclusionary to each other AND use all ten digits just once each.
I suppose there are lots of numbers that generate squares and cubes that are merely exclusionary to each other? What is the first three digit example, four digit example? ... ?
Is it possible to construct the fraction 1/2 by summing other fractions of the form 1/x2? For example, you can choose various denominators as in: 1/32 + 1/52 + 1/102 + ... (but this is not an answer!) The solution must have a finite number of terms. 1) Fragile Fractions of the 1st Kind: No value of x may be greater than 100. No values of x may be repeated. 2) Fragile Fractions of the 2nd Kind: you may use values greater than 100. Once you have found a solution for 1/2, can you construct a solution for: 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9?
1/2 = (1/2)^2+(1/3)^2+(1/4)^2+(1/5)^2+(1/6)^2+(1/12)^2+ (1/30)^2+(1/60)^2+(1/75)^2+(1/100)^2 1/3 = (1/2)^2+(1/4)^2+(1/8)^2+(1/24)^2+(1/28)^2+(1/30)^2+ (1/40)^2+(1/56)^2+(1/84)^2 1/4 = (1/3)^2+(1/4)^2+(1/5)^2+(1/6)^2+(1/12)^2+(1/30)^2+ (1/60)^2+(1/75)^2+(1/100)^2 1/5 = (1/3)^2+(1/4)^2+(1/7)^2+(1/20)^2+(1/28)^2+(1/30)^2+ (1/35)^2+(1/60)^2 1/6 = (1/3)^2+(1/5)^2+(1/9)^2+(1/18)^2+(1/90)^2 1/7 = (1/3)^2+(1/6)^2+(1/20)^2+(1/42)^2+(1/60)^2+(1/70)^2+ (1/75)^2+(1/84)^2+(1/100)^2 1/8 = (1/3)^2+(1/9)^2+(1/36)^2+(1/45)^2+(1/60)^2 1/9 = (1/4)^2+(1/5)^2+(1/12)^2+(1/30)^2+(1/60)^2+(1/75)^2+ (1/100)^2 The are also two trivial solutions: 1/4 = (1/2)^2 and 1/9 = (1/3)^2 Ilan Mayer, Toronto, Canada (ilan at cedara dot com) David Jones
says If were were allowed to have numbers repeat in the denominator, the problem would be trivial 1/2 = 1/(2^2) + 1/(2^2) = 1/(4^2) + 1/(4^2) + repeated 8 times. I'm sure you can see a pattern forming. For the first problem, the difficutly I think comes in the fact that in order to get to a fraction like 1/2 is that it is difficult to introduce another denominator with a prime factor that is not 2. For example, let say that 1/(3^2) is your first term. Well, there is no way to add more ninths terms that get you back to 1/2. You can't cancel either of those threes out of the demoninator (even if you let x go over 100). So, the best you can do is only use terms that are powers of two. But we know that 1/2 = 1/4 + 1/8 + 1/16 + 1/32 + etc and by the rules for the problem some of these items are missing (like 1/8th) so we know that we can never get there even we were allowed an infinite number of terms. This is very poorly contructed arguement, so I will not claim this as a fact, but my supposition is that such a set of fractions that sum to 1/2 does not exist. Davy
> Would you press the button to save a loved > one from certain, imminent death?Much harder question. I should say no for a variety of ethical reasons, but it's a lot easier to see what you gain personally out of such a bargain. I can't honestly say that I would have the fortitude not to push it, espically if by "imminent" you mean that I don't have more than a few minutes to ponder the situation. > Would you press the button to save a huge rain > forest from imminent destruction -- say the rain > forest would be guaranteed to persist for a > century? No. Going back to the karma arguement, I'm letting somebody off the hook for destroying the forest and accepting the blame for a death. Much like the stoning, there is no guarantee that they just won't go do it to a different rain forest or other valuable natural resource. And what after 100 years? If we still haven't learned how to respect our ecosystem it will be gone then. See, the deal has involve some sort lost lasting educational benefit to all of humanity before it becomes worthwhile. In situations like this, its not enough to just stop people; they have to understand why its important to stop. Besides, if humanity is so stupid as destroy a major source of our oxygen supply, maybe the planet would be better off if we all became extinct as a result of it. Nature manages in its own way.
> Would you press the button to save the current U.S. > President?I could give quite a few reasons why I would be tempted to NOT push the button, but I appreciate the fact that this group has never become overly political and I'd like to keep it that way, so I'll keep this one under my hat. But even if it were a President that I valued, still probably not.
> Would you press the button to save yourself?See "loved one"
> Next, consider the same scenarios, except that the > person, selected at random, is from the town you > live in.Wouldn't change anything.
> I'll assume I can mention your ideas in a book, with > credit (for example, I can refer to you by first > name and last initial), unless you tell me not to.Heck, I'd actually prefer David C. Jones so I could tell all my friends "Look everybody, I'm in a book!"
Reply (as a sonnet couple) If I this Button before me now press, Someone unbeknownst to me perish, Yet someone saved whom I dearly cherish, For Greater Good this Act -- oh what distress To make such choice! And yet, is this choice less Ethical than others make when they wish To rid the world of evil with their harsh Wars and soldiers, who to their duties pass To service of a greater call? To halt My hand in uneasy ethic to save Some unknown at the cost of one I love, At this my conscience would later find fault, And thus, I'd press and pray forgiveness brave, Rather than some notion of ethics prove. II To end the madness that is stoning death, To save a multitude from fate rock hit, Again, I'd play at being God, unfit Though I am, even for a single breath, To pluck that random price from all the wealth Of those who would thus be from my choice hit, And though some random martyr pay for it, I'd ask forgiveness for my act uncouth. O fickle Button! Why do you now place Such power over life and death on me? I am not worthy to yield such Decide, But since no other has fall'n to my place, I'll press you, damn it, though respectfully, And pray my decision I may yet abide. -- Quinn Tyler Jackson (if you use it, credit me as you will)
> 5. Before you is a red button. Press it, and one person, selected > at random, dies. However, as a direct, instantaneous result of your > pressing the button, the stoning of suspected adulteresses in Islamic > countries halts forever. Do you press the button?This variation of this question has been asked already: "Before you is a red button. Press it, and ten-million prople, selected at random, die. However, as a direct, instantaneous result of your pressing the button, the form of government known as communism halts forever.
> Would you press > the button to save a loved one from certain, imminent death? Would > you press the button to save a huge rain forest from imminent > destruction -- say the rain forest would be guaranteed to persist for > a century? Would you press the button to save the current U.S. > President? Would you press the button to save yourself? Next, > consider the same scenarios, except that the person, selected at > random, is from the town you live in.Since I don't believe in magic, I always wonder how exactly the button would work. I think there would be much profit to be made if one could invent a button that killed people at a distance.
5. Before you is a red button. Press it, and one person, selected at random, dies. However, as a direct, instantaneous result of your pressing the button, the stoning of suspected adulteresses in Islamic countries halts forever. Do you press the button?No, let nature take its' course
> Would you press > the button to save a loved one from certain, imminent death?Probably, after all who wouldn't?
>Would you press the button to save a huge rain forest from imminent > destruction -- say the rain forest would be guaranteed to persist for a century?No, not worth the human life.
Would you press the button to save the current U.S. President?Not if it meant someone else arbitrarily and undeservedly dies. What that amounts to is trading one life for another. No.
> Would you press the button to save yourself?Same as above re loved one, yes, human nature prevails.
> Next, consider the same scenarios, except that the person, selected at > random, is from the town you live in.No change
· Would you press the button to save a loved one from certain, imminent death? Absolutely. · Would you press the button to save a huge rain forest from imminent destruction – say the rain forest would be guaranteed to persist for a century? Survival of the rain forest would have an impact on more than just one person, so yes. · Would you press the button to save the current U.S. President? I would be gambling on the possibility that the randomly chosen person might have more long-term positive influence than a barely elected good-ole-boy who can barely utter a coherent sentence (besides, he has enough brainpower surrounding him that his removal would barely be registered). No. · Would you press the button to save yourself? Yes. (Is an explanation really necessary?) · Next, consider the same scenarios, except that the person, selected at random, is from the town you live in. The initial "one person, selected at random" to me implied anyone on Earth – meaning that the odds of a friend or family member's being selected were exceedingly low. However, limiting the choice to a population of a few hundred thousand greatly increases those odds … so No.
David writes: In the problem, there are three types of objects: Alloys, Gems, and Spices. Each type has four colors: Beige, Green, Pink, and Yellow. Each item will be referred to with two letters, the first by color and then second by type. For example, Pink Spices will be annoted by PS, Beige Alloys are annotated by BA, and so on.
The second paragraph of the problem talks exclusively about gems. The problem will be started and dealing with alloys and spices will come later. An elementary interpretation of the paragraph yields the following equations, labeled A, B, and C.
Algebraic manipulation allows us to rewrite these equations as:
Color Type Variable Amount Blue Gems BG 98280 Green Gems GG 63840 Pink Gems PG 121170 Yellow Gems YG 135760 Blue Alloys BA 871035 Green Alloys GA 1045242 Pink Alloys PA 4802600 Yellow Alloys YA 508332 Blue Spices BS 105 Green Spices GS 6 Pink Spices PS 484710 Yellow Spices YS 484809 Total 8615889
> [Cliff says, "New York City has the best doctors in the world. > If I had to get sick, it would be in New York.Probably true, but what if the only treatment and diagnostic equipment for your illness was offered at the Mayo Clinic in Minnesota? Myself, I would hate to be restricted.
--- PedersenOk, but what about less freakish events? What happens when my best friend gets married? Can't go to the wedding unless I'm willing to pay to fly the whole party to NYC (that is, if the bride and groom are willing). When my dad is on his death bed, am I going to be there? When both mom and dad die how will I take care of the estate? Ok, I guess for 40 million I can move the family out to live with me before any of that happens. I've always wanted to go scuba diving off some tropical reef where I can swim with the fish. That one's out the window for sure. (I suppose not if they have an aquarium, but what kind of grant would I have to give for a dip in the tank?) I'm definitely never going to get to see the Pyramids of Egypt which I've wanted to do for most of my life. For me personally, I have asthma. I don't know that the higher pollution levels would good for me and no amount of money is going to cure that. I think the main thing is that being just under 30, I have too much life left to life. 40 million dollars probably comes out just enough to comp for all the things I would miss doing, miss out on never being able to do, and the expenses of transporting the valuable things of my life with me. Make the offer again in, say, 20 or 30 years and I would probably take it.
wrote: > * Ha. But freakish events can strike anywhere at any > time. Might as well have fun spending the 40 million > carpe diem style!
11 is an isoprime, a prime number with all digits the same. Do any other isoprimes exist? 101 is an oscillating bit prime. Do any others exist? For example, 10101 is not prime. Neither is 1010101. Cliff www.pickover.com
SolutionsDavid Jones: If any more exist, they would also have to be all ones. If you take any other number, say repeating 3's, the number is automatically divisible by 3. Furthermore, the number of 1s in the number would have to be prime itself. Say I string together 15 1s. Since 3 and 5 divide 15, I can quickly conclude that this number is divisible by 111 or 11111. [Cliff says, "Do you think humanity will ever find a 1111... isoprime?"]> 101 is an oscillating bit prime. Do any others > exist? For example, 10101 is not prime. Neither is > 1010101.Much like the example above, we know that such primes will have to end with the same number they start with. For example, 737373737373 can't be prime since 73 divides it. We need to add a seven to the end to make it a viable candidate. I don't have any other brilliant leads beyond this.
I seem to recall a few months ago that you proposed other people come up with special classes of numbers. A few suggestions were made about "heavenly primes" and the like. If memory serves, some of them were oscillating primes and a handful of them were found. Davy
says: According to The Penguin Dictionary of Curious and Interesting Numbers, the next isoprime is 1,111,111,111,111,111,111. The third one is 11,111,111,111,111,111,111,111.
David Jones: Well, the first couple aren't that hard to find. Define a function ISO(n)= n number of ones. For example, ISO(2)=11, ISO(3)=111, ISO(4)=1111 and so on. By toying with Maple I found that ISO(19) and ISO(23) are both prime which I verified with my 49G. I think I tested it up to about ISO(83) before I put it down. I could have easily made an error as I was entering numbers by hand and may have miscounted digits. I don't know Maple well enough yet that I could write a script for it to create and test numbers for me and PrimeForm won't let me define functions like that. Maybe tommorrow at work I'll play with it a bit more.
says: In the world of factoring and primality proving 11 is considered a repunit prime. All repunit primes in base 10 can only be composed of 1's. the others known have 19 digits, 23 digits, 317 digits, and 1031 digits. The next 2 which are believed to be prime but are not proven such yet are 49081 digits and 86453 digits. For any info regarding prime numbers there is a great web site run by Chris Caldwell at http://primes.utm.edu/
Daniel Dockery: Unless I err, the fourth isoprime would be:11,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111,111,111,111,111, 111,111,111,111,111,111(317 digits) Both Primo and Maple certify the number as prime. At 20020127T0850, Cliff wrote:> 101 is an oscillating bit prime. Do any others exist? > For example, 10101 is not prime. Neither is 1010101.I've found no others less than 10**700 (the highest I've searched so far). Daniel
David Jones: Just to give you an idea of how hard it is to find these things (isoprimes), lets go back to our problem of Specially Augmented (aka "Jim") Primes from a few months ago. Jim Primes and Repunit primes grow at about the same rate for N. On November 4th I complete proving that n=3,011 was prime. At the same time, I had a separate computer looking for more and it was at n=23,000. Today, 85 days later, I am testing n=47,461. My N value currently goes up by about a 1,000 every week.
Using a log(log(n)) fit to predict where the next SAP should be, it should have been at about 39,000 but it didn't appear. The next one should be at around n=245,000. I haven't quite figured out a reliable timetable yet, but an optimistic prediction shows that I won't get there until about April of next year unless I either get a faster computer, a faster testing program, or more help.
If/When I do find one, it will only have passed a preliminary test for "probable" primality. Doing a full primality proof on n=3011 took a couple of weeks if I remember correctly. I would imagine that proving a prime on n=245,000 would take at least three months. The individual proving ISO(86,453) has my best wishes as I would hate to lose months of computing time just to find out my candidate is NOT a prime number.
The moral of the story here though is that, now days, beating any sort of prime record requires a ton of patience and even more persistence. Davy
Daniel Dockery:At 200201281158, David Jones wrote:> The individual proving ISO(86,453) has my best wishes > as I would hate to lose months of computing time just > to find out my candidate is NOT a prime number.On a first run, PFGW yielded a 3-prp result for the 49081 digit item after about 10 minutes (with thread priority; all the other runs were set for only idle cycles), and a "composite" result for the 86,453 digit item after about 45 minutes. A second run repeated 3-prp for 49081, but yielded 3-prp for 86453 this time, both with about the same speed. Running a primality test (pfgw -t -f -l) cycled through, I think, base 17 before declaring 49081 composite; time was close to 30 minutes. I haven't yet attempted the full test on 86453, and may not at this point. Daniel
" Cliff wrote:> > 11 is an isoprime, a prime number with all digits the same. Do any other > isoprimes exist?It's conventionally called a "repunit", standing for "repeated unit", i.e. string of 1s. A list of the largest known repunits can be found on Professor Caldwell's Prime Pages http://primepages.org/In particular: http://primes.utm.edu/glossary/page.php/Repunit.html > 101 is an oscillating bit prime (OBP). Do any others exist? For example, > 10101 is not prime. Neither is 1010101. Humans have not disovered any > other OBPs, as far as I know.What base is that in? You mention 'bit', so should I assume it's base 2? In which case, these are simply the repunits in base 4. If it's base 10, then these are simply the replunits in base 100. Repunits in bases other than ten are typically called Generalised Repunits. In base 10, primes with the generalisation of this pattern are typically called "undulating". There have been some very famous undulating primes, such as Landon Curt Noll's '37' one. Rudolf Ondrejka's Top Ten lists may well contain some of your '101...01' type , it contains all sorts of things. http://www.utm.edu/research/primes/lists/top_ten/> After you read through the discussion on this subject at: > http://sprott.physics.wisc.edu/pickover/extremec.html > > you'll discover that the largest isoprime published in a book is: > > 11,111,111,111,111,111,111,111.Book? What kind of 16th century medium is that? The Cunningham Project has been finding factors of numbers of such form (and thus finding primes too) seemingly for ever, well over a decade. http://www.cerias.purdue.edu/homes/ssw/cun/> And, we think we have discovered a new one: > > 11,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111,111,111,111,111, > 111,111,111,111,111,111 > > My question: do you think humanity will ever discover a larger > isoprime? Upon what do you base your reasoning?You've obviously never heard of Harvey Dubner! (Having said that it wasn't Harvey but Lew Baxter who found a probable repunit prime with 86453 digits only 15 months ago.) The largest known _proved_ repunit is the 1031-digit one found to be PRP by Harvey Dubner some time in the mists of time, but proved to be actually prime only more recently. Phil
Jan Kristian Haugland
:George Weinberg wrote: > On Wed, 30 Jan 2002 08:40:04 -0500, Cliff>Phil Carmody > wrote: > >11 is an isoprime, a prime number with all digits the same. Do any other > >isoprimes exist? > > Funny thing, when I first read this I somehow got the > impression you were talking in binary, maybe it's > the oscillatong but thing. 11 seems to be prime in quitre a lot of > bases. Yes, it "seems" to be prime whenever the base is a prime minus one. ;-) (...) > >My question: do you think humanity will ever discover a larger > >isoprime? Upon what do you base your reasoning? > > > > No. Didn't read the discussion, so I may be wrong, but I think > that the frequency of isonumbers falls off fast enough and the > frequency of primes falls off fast enough that even going out to > infinity there's likely only a finite number. The frequency of primes falls off, but not very fast, so it is not unreasonable that there may be infinitely many. That however doesn't necessarily mean humanity will find too many others. Compare with Mersenne primes. wrote:> >>> My question: do you think humanity will ever discover a larger >>> isoprime? Upon what do you base your reasoning? > >> Facts and research. > >In 1993 Dr. Ron Rivest did a calculation on the number of MIPS-years >it takes to factor a number (it's in the appendix of my O'Reilly PGP >book, dunno if the calculation has been updated). You could probably takethe iso-prime research and run it through his calculator to determine >when humanity could find the next isoprime. You'll need to take into account improvements in factoring -- I believe there have been several since 1993 and I see no reason to believe there won't be more. -- Matthew T. Russotto email@example.com --