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| '''Singmaster's conjecture''' is a [[conjecture]] in [[combinatorial number theory]] in [[mathematics]], named after the British professor [[David Singmaster]] who proposed it in 1971. It says that there is a finite [[upper bound]] on the [[multiplicity (mathematics)|multiplicities]] of entries in [[Pascal's triangle]] (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in [[Pascal's triangle]] is 1, because any other number ''x'' can appear only within the first ''x'' + 1 rows of the triangle. [[Paul Erdős]] said that Singmaster's conjecture is probably true but he suspected it would be very hard to prove.
| | The name of the author is Figures. Hiring is my profession. What I love doing is performing ceramics but I haven't produced a dime with it. Puerto Rico is exactly where he and his wife reside.<br><br>Take a look at my weblog ... at home std testing ([http://www.Ninfeta.tv/blog/99493 index]) |
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| Let ''N''(''a'') be the number of times the number ''a'' > 1 appears in Pascal's triangle. In [[big O notation]], the conjecture is:
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| :<math>N(a) = O(1).\,</math>
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| ==Known results==
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| Singmaster (1971) showed that
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| :<math>N(a) = O(\log a).\,</math>
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| Abbot, [[Paul Erdős|Erdős]], and Hanson (see '''References''') refined the estimate. The best currently known (unconditional) bound is
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| :<math>N(a) = O\left(\frac{(\log a)(\log \log \log a)}{(\log \log a)^3}\right),\,</math>
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| and is due to [[Daniel Kane (mathematician)|Kane]] (2007). Abbot, Erdős, and Hanson note that conditional on [[Cramér's conjecture]] on gaps between consecutive primes that
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| :<math> N(a) = O\left( \log(a)^{2/3+\varepsilon}\right) </math>
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| holds for every <math>\varepsilon > 0 </math>.
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| Singmaster (1975) showed that the [[Diophantine equation]]
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| :<math>{n+1 \choose k+1} = {n \choose k+2},</math>
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| has infinitely many solutions for the two variables ''n'', ''k''. It follows that there are infinitely many entries of multiplicity at least 6. The solutions are given by
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| :<math>n = F_{2i+2} F_{2i+3} - 1,\,</math>
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| :<math>k = F_{2i} F_{2i+3} - 1,\,</math>
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| where ''F''<sub>''n''</sub> is the ''n''th [[Fibonacci number]] (indexed according to the convention that ''F''<sub>1</sub> = ''F''<sub>2</sub> = 1).
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| ==Numerical examples==
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| Computation tells us that
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| * 2 appears just once; all larger positive integers appear more than once;
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| * 3, 4, 5 each appear 2 times;
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| * all odd prime numbers appear 2 times;
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| * 6 appears 3 times;
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| * Many numbers appear 4 times.
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| * Each of the following appears 6 times:
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| :: <math>{120 \choose 1} = {16 \choose 2} = {10 \choose 3}</math>
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| <!-- deliberately added extra space -->
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| :: <math>{210 \choose 1} = {21 \choose 2} = {10 \choose 4}</math>
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| <!-- deliberately added extra space -->
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| :: <math>{1540 \choose 1} = {56 \choose 2} = {22 \choose 3}</math>
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| <!-- deliberately added extra space -->
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| :: <math>{7140 \choose 1} = {120 \choose 2} = {36 \choose 3}</math>
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| <!-- deliberately added extra space -->
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| :: <math>{11628 \choose 1} = {153 \choose 2} = {19 \choose 5}</math>
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| <!-- deliberately added extra space -->
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| :: <math>{24310 \choose 1} = {221 \choose 2} = {17 \choose 8}</math>
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| * The smallest number to appear 8 times is 3003, which is also the first member of Singmaster's infinite family of numbers with multiplicity at least 6:
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| :: <math>{3003 \choose 1} = {78 \choose 2} = {15 \choose 5} = {14 \choose 6}</math>
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| The next number in Singmaster's infinite family, and the next smallest number known to occur six or more times, is 61218182743304701891431482520.
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| It is not known whether any number appears more than eight times, nor whether any number besides 3003 appears that many times. The conjectured finite upper bound could be as small as 8, but Singmaster thought it might be 10 or 12.
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| ===Do any numbers appear exactly five or seven times?===
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| It would appear from a related entry, {{OEIS|id=A003015}} in the [[Online Encyclopedia of Integer Sequences]], that no one knows whether the equation ''N''(''a'') = 5 can be solved for ''a''. Nor is it known whether any number appears seven times.
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| ==See also==
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| * [[Binomial coefficient]]
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| ==References==
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| *{{citation
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| | last = Singmaster | first = D. | author-link = David Singmaster
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| | doi = 10.2307/2316907
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| | mr = 1536288
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| | issue = 4
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| | journal = [[American Mathematical Monthly]]
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| | pages = 385–386
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| | title = Research Problems: How often does an integer occur as a binomial coefficient?
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| | volume = 78
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| | year = 1971
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| | jstor = 2316907}}.
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| *{{citation
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| | last = Singmaster | first = D. | author-link = David Singmaster
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| | mr = 0412095
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| | issue = 4
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| | journal = [[Fibonacci Quarterly]]
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| | pages = 295–298
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| | title = Repeated binomial coefficients and Fibonacci numbers
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| | url= http://www.fq.math.ca/Scanned/13-4/singmaster.pdf
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| | volume = 13
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| | year = 1975}}.
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| *{{citation
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| | last1 = Abbott | first1 = H. L.
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| | last2 = Erdős | first2 = P. | author2-link = Paul Erdős
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| | last3 = Hanson | first3 = D.
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| | doi = 10.2307/2319526
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| | mr = 0335283
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| | journal = [[American Mathematical Monthly]]
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| | pages = 256–261
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| | title = On the number of times an integer occurs as a binomial coefficient
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| | volume = 81
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| | year = 1974
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| | jstor = 2319526
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| | issue = 3}}.
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| *{{citation
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| | last = Kane | first = Daniel M. | author-link = Daniel Kane (mathematician)
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| | mr = 2373115
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| | journal = [[Integers: Electronic Journal of Combinatorial Number Theory]]
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| | url = http://www.emis.de/journals/INTEGERS/papers/h53/h53.pdf
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| | pages = #A53
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| | title = Improved bounds on the number of ways of expressing ''t'' as a binomial coefficient
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| | volume = 7
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| | year = 2007}}.
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| ==External links==
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| * {{OEIS|id=A003016}} (OEIS = [[Online Encyclopedia of Integer Sequences]])
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| [[Category:Combinatorics]]
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| [[Category:Number theory]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Triangles of numbers]]
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| [[Category:Conjectures]]
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The name of the author is Figures. Hiring is my profession. What I love doing is performing ceramics but I haven't produced a dime with it. Puerto Rico is exactly where he and his wife reside.
Take a look at my weblog ... at home std testing (index)