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

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An Interesting Note
10 is both a triangular number and a tetrahedral
number.
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Why does this hold?
Is it equal to any other binomial coefficient?
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Are there any non-trivial relations?
Yields a Pell Equation
with Solutions
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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)

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What are Reasonable Bounds?

• Singmaster showed in 2 that
• Consider n gt 2m (we do this from now on)

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An improvement

• In 3 Erdös et al. proved that
• Prime in (n-n5/8, n) for n gtgt 1.
• Split into cases based on n gt (log t)6/5

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For n gt (log t)6/5

• Use Approximation

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For n lt (log t)6/5

• Approximation yields m gt n5/8
• 9 prime, P, s.t. n-m lt P lt n
• P divides t
• Pick largest N, all others satisfy P lt n lt N
• At most M solutions
• M O(N5/8) O((log t)3/4)
• Using the strongest conjectures on gaps between
  primes could give

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Another way to handle this case

• Consider all solutions
• Order so n1 gt n2 gt gt nk
• m1 lt m2 lt lt mk,
• As n decreases and m increases, the effect of
  changing n decreases and the effect of changing m
  increases.
• (ni-ni1)/(mi1-mi) increasing.
• These fractions are distinct

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Another way (continued)

• Differences of (ni , mi) are distinct
• Only O(s2) have ?n, ?m lt s
• Total Change in n-m gt ck3/2
• k3/2 O((log t)6/5)
• k O ((log t)4/5).

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What did we do?

• n convex function of m
• Lattice points on graph of ANY convex function
• Idea Consider higher order derivatives.

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Problems to Overcome

• How do we use the derivative data?
• How do we obtain the derivative data?

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Using Data, a Lemma

• Lemma If f and g are Cn-functions so that f(x)
  g(x) at x1 lt x2 lt lt xn1, then
• f(n)(y) g(n)(y) for some y2 (x1 , xn1)
• Generalization of Rolles Theorem
• Proof by induction Rolles Theorem

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Using Data, Proof of Lemma
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Using Derivative Data (Continued)

• Consider function f, f(mi) ni
• g polynomial interpolation of f at m1, m2, ,
  mk1
• g(k) constant
• f(k) is this constant at some point

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Using Derivative Data (cont.)
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Using Derivative Data (cont.)

• kth derivate of f small but non-zero
• Fit polynomial to k1 points separated by S
• ) kth derivative over k! either 0 or more than
  S-k(k1)/2.

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Derivative Data

• Define
• Make smooth using ?
• Estimate with Sterlings approximation

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Estimating Derivatives

• Main Term Derivatives of
• log t ?(z1) / z and take exp of power
  series
• (z - 1) / 2 is easy
• Error term Cauchy Integral Formula

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A Useful Parameter
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Split into Cases

• ? lt 1.15
• 1.15 lt ? lt log log t/(24 log log log t)
• log log t/(24 log log log t)lt ? lt(log log t)4
• (log log t)4 lt ?

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Case 1 ? lt 1.15

• Already covered
• O((log t)3/4) solutions

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Case 2 1.15 lt ? lt log log t/(24 log log log t)

• Set k (log log t)/(12 log log log t)
• Technical conditions satisfied
• k1 adjacent solutions mi of separation S

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Case 3 log log t/(24 log log log t)lt ? lt(log
log t)4

• We need a slightly better analysis to bound

LCM over all sequences of k distinct ri? 0, ri
lt S
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Bounding B

• Count multiples of each prime
• pn divides at most min(k , 2S/pn) ris
• Use Prime Number Theorem

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Using Bound

• k 2?
• Technical Conditions Satisfied

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Case 4 (log log t)4 lt ?
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Conclusions

• Know where to look to tighten this bound
• Can use technique for other problems

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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.