the number of ways of expressing t as a binomial coefficient l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The Number of Ways of Expressing t as a Binomial Coefficient PowerPoint Presentation
Download Presentation
The Number of Ways of Expressing t as a Binomial Coefficient

Loading in 2 Seconds...

play fullscreen
1 / 30

The Number of Ways of Expressing t as a Binomial Coefficient - PowerPoint PPT Presentation


  • 500 Views
  • Uploaded on

The Number of Ways of Expressing t as a Binomial Coefficient By Daniel Kane January 7, 2007 An Interesting Note: 10 is both a triangular number and a tetrahedral number. Why does this hold? Is it equal to any other binomial coefficient? Trying to produce solutions of the form

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The Number of Ways of Expressing t as a Binomial Coefficient' - jaden


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
an interesting note
An Interesting Note:

10 is both a triangular number and a tetrahedral number.

why does this hold
Why does this hold?

Is it equal to any other binomial coefficient?

stating the problem more precisely

Define

Stating the Problem More Precisely
  • Studied by Singmaster in [2] (1971)
  • N(3003) = 8
  • N(t) ¸ 6 for infinitely many t
  • Conj: N(t) = O(1)
what are reasonable bounds
What are Reasonable Bounds?
  • Singmaster showed in [2] that
  • Consider n > 2m (we do this from now on)
an improvement
An improvement
  • In [3] Erdös et al. proved that
  • Prime in (n-n5/8, n) for n >> 1.
  • Split into cases based on n > (log t)6/5
for n log t 6 5
For n > (log t)6/5
  • Use Approximation
for n log t 6 59
For n < (log t)6/5
  • Approximation yields m > n5/8
  • 9 prime, P, s.t. n-m < P < n
  • P divides t
  • Pick largest N, all others satisfy P < n < N
  • At most M solutions
  • M = O(N5/8) = O((log t)3/4)
  • Using the strongest conjectures on gaps between primes could give
another way to handle this case
Another way to handle this case
  • Consider all solutions
  • Order so n1 > n2 > … > nk

m1 < m2 < … < mk,

  • As n decreases and m increases, the effect of changing n decreases and the effect of changing m increases.
  • (ni-ni+1)/(mi+1-mi) increasing.
  • These fractions are distinct
another way continued
Another way (continued)
  • Differences of (ni , mi) are distinct
  • Only O(s2) have n, m < s
  • Total Change in n-m > ck3/2
  • k3/2 = O((log t)6/5)
  • k = O ((log t)4/5).
what did we do
What did we do?
  • n convex function of m
  • Lattice points on graph of ANY convex function
  • Idea: Consider higher order derivatives.
problems to overcome
Problems to Overcome
  • How do we use the derivative data?
  • How do we obtain the derivative data?
using data a lemma
Using Data, a Lemma

Lemma If f and g are Cn-functions so that f(x) = g(x) at x1 < x2 < … < xn+1, then

f(n)(y) = g(n)(y) for some y2 (x1 , xn+1)

  • Generalization of Rolle’s Theorem
  • Proof by induction & Rolle’s Theorem
using data proof of lemma

y

f(n)(x) = g(n)(x)

f(2)(x) = g(2)(x)

f(1)(x) = g(1)(x)

f(0)(x) = g(0)(x)

x1

x2

x3

xn-1

xn

xn+1

Using Data, Proof of Lemma
using derivative data continued
Using Derivative Data (Continued)
  • Consider function f, f(mi) = ni
  • g polynomial interpolation of f at m1, m2, …, mk+1
  • g(k) constant
  • f(k) is this constant at some point
using derivative data cont18
Using Derivative Data (cont.)
  • kth derivate of f small but non-zero
  • Fit polynomial to k+1 points separated by S
  • ) kth derivative over k! either 0 or more than S-k(k+1)/2.
derivative data
Derivative Data
  • Define
  • Make smooth using 
  • Estimate with Sterling’s approximation
estimating derivatives
Estimating Derivatives
  • Main Term: Derivatives of

[ log t + (z+1) ] / z and take exp of power series

  • (z - 1) / 2 is easy
  • Error term: Cauchy Integral Formula
split into cases
Split into Cases
  •  < 1.15
  • 1.15 <  < log log t/(24 log log log t)
  • log log t/(24 log log log t)<  <(log log t)4
  • (log log t)4 < 
case 1 1 15
Case 1:  < 1.15
  • Already covered
  • O((log t)3/4) solutions
case 2 1 15 log log t 24 log log log t
Case 2: 1.15 <  < log log t/(24 log log log t)
  • Set k = (log log t)/(12 log log log t)
  • Technical conditions satisfied
  • k+1 adjacent solutions mi of separation S
case 3 log log t 24 log log log t log log t 4
Case 3: log log t/(24 log log log t)<  <(log log t)4

We need a slightly better analysis to bound

LCM over all sequences of k distinct ri 0,

|ri| < S

bounding b
Bounding B
  • Count multiples of each prime
  • pn divides at most min(k , 2S/pn) ri’s
  • Use Prime Number Theorem
using bound
Using Bound
  • k = 2
  • Technical Conditions Satisfied
conclusions
Conclusions
  • Know where to look to tighten this bound
  • Can use technique for other problems
references
References

[1] D. Kane, On the Number of Representations of t as a Binomial Coefficient, Integers: Electronic Journal of Combinatorial Number Theory,4, (2004), #A07, pp. 1-10.

[2] D. Singmaster, How often Does an Integer Occur as a Binomial Coefficient?, American Mathematical Monthly, 78, (1971) 385-386

[3] H. L. Abbott, P. Erdös and D. Hanson, On the number of times an integer occurs as a binomial coefficient, American Mathematical Monthly, 81, (1974) 256-261.