The Number of Ways of Expressing t as a Binomial Coefficient

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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

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### 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?

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?
• Singmaster showed in [2] that
• Consider n > 2m (we do this from now on)
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
• Use Approximation
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
• 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)
• 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?
• n convex function of m
• Lattice points on graph of ANY convex function
• Idea: Consider higher order derivatives.
Problems to Overcome
• How do we use the derivative data?
• How do we obtain the derivative data?
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

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)
• 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 (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
• Define
• Make smooth using 
• Estimate with Sterling’s approximation
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
•  < 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
• O((log t)3/4) solutions
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

We need a slightly better analysis to bound

LCM over all sequences of k distinct ri 0,

|ri| < S

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