Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal's Identity

@article{MacMillan2011ProofsOP,
  title={Proofs of Power Sum and Binomial Coefficient Congruences Via Pascal's Identity},
  author={Kieren MacMillan and Jonathan Sondow},
  journal={The American Mathematical Monthly},
  year={2011},
  volume={118},
  pages={549 - 551}
}
Abstract A well-known and frequently cited congruence for power sums is where n ≥ 1 and p is prime. We survey the main ingredients in several known proofs. Then we give an elementary proof, using an identity for power sums proven by Pascal in 1654. An application is a simple proof of a congruence for certain sums of binomial coefficients, due to Hermite and Bachmann. 

A Proof of Symmetry of the Power Sum Polynomials Using a Novel Bernoulli Number Identity

TLDR
An elementary proof that the sum of p-th powers of the first natural numbers can be expressed as a polynomial in n of degree $p+1$ is given and a novel identity involving Bernoulli numbers is proved.

An Explicit Identity to solve sums of Powers of Complex Functions

A recurrence relations for sums of powers of complex functions can be written as a system of linear equation AX=B. Using properties of determinant and Cramer's rule for solving systems of linear

Power-Sum Denominators

TLDR
Asquarefree product formula for the denominators of the Bernoulli polynomials is derived and it is shown that such a denominator equals n + 1 times the squarefree product of certain primes p obeying the condition that the sum of the base-p digits of n +1 is at least p.

Reducing the Erdős–Moser Equation 1 n + 2 n + ⋯ + kn = (k + 1) n Modulo k and k 2

TLDR
Reducing the equation modulo k and k 2, it is given necessary and sufficient conditions on solutions to the resulting congruence and supercongruence to prove the conjecture that the only solution of the Diophantine equation is the trivial solution 1 + 2 = 3.

On a Congruence Modulo n 3 Involving Two Consecutive Sums of Powers

For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k +2 k +···+n k , are some of the most popular sums in all of mathematics. In this note we prove a

Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series

In a recent paper the author derived a formula for calculating common denominators for the homogeneous components of the Baker–Campbell-Hausdorff (BCH) series. In the present work it is proved that

Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series

In a recent paper the author derived a formula for calculating common denominators for the homogeneous components of the Baker-Campbell-Hausdorff (BCH) series. In the present work it is proved that

A congruence modulo $n^3$ involving two consecutive sums of powers and its applications

For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n+ 1) = 1 k + 2 + · · ·+ n, have got to be some of the most popular sums in all of mathematics. In this

Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality

The Tijdeman-Zagier conjecture states no integer solution exists for AX + BY = CZ with positive integer bases and integer exponents greater than 2 unless gcd(A,B,C) > 1. Any set of values that

Solutions of the congruence 1+2f(n)+⋯+nf(n)≡0 mod n

In this paper we characterize, in terms of the prime divisors of n, the pairs (k,n) for which n divides ∑j=1njk . As an application, we derive some results on the sets Mf:=n≥1:f(n)>1 and ∑j=1njf(n)≡0

References

SHOWING 1-10 OF 29 REFERENCES

Congruences for a class of alternating lacunary sums of binomial coefficients

An 1876 theorem of Hermite, later extended by Bachmann, gives congruences modulo primes for lacunary sums over the rows of Pascal’s triangle. This paper gives an analogous result for alternating sums

The Equivalence of Giuga's and Agoh's Conjectures

In this paper we show the equivalence of the conjectures of Giuga and Agoh in a direct way which leads to a combined conjecture. This conjecture is described by a sum of fractions from which all

History of the Theory of Numbers

THE third and concluding volume of Prof. Dickson's great work deals first with the arithmetical. theory of binary quadratic forms. A long chapter on the class-number is contributed by Mr. G. H.

A Top Hat for Moser's Four Mathemagical Rabbits

TLDR
It is shown here that Moser's result can be derived from a von Staudt-Clausen type theorem and the mathematical arguments used in the proofs were already available during the lifetime of Lagrange.

Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k

If the equation of the title has an integer solution with k>=2, then m>10^{10^6}. Leo Moser showed this in 1953 by amazingly elementary methods. With the hindsight of more than 50 years his proof can

Pascal's arithmetical triangle

Imagine having some marbles, pebbles, or other objects that you want to lay out in a neat triangular pattern. How many do you need to end up with a complete triangle? Three will do; so will 6, 10,

An Introduction to the Theory of Numbers

  • E. T.
  • Mathematics
    Nature
  • 1946
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.

Sommation des puissances numériques

  • Oeuvres complètes, vol. III, Jean Mesnard, ed., Desclée-Brouwer, Paris, 1964, 341–367; English translation by A. Knoebel, R. Laubenbacher, J. Lodder, and D. Pengelley, Sums of numerical powers, in Mathematical Masterpieces: Further Chronicles by the Explorers, Springer Verlag, New York
  • 2007

Number Theory: Volume II: Analytic and Modern Tools