On the Brocard–Ramanujan Diophantine Equation n! + 1 = m2

In 1876, H. Brocard posed the problem of finding all integral solutions to n! + 1 = m². In 1913, unaware of Brocard's query, S. Ramanujan gave the problem in the form, The number 1 + n! is a perfect square for the values 4, 5, 7 of n. Find other values. We report on calculations up to n = 10⁹ and briefly discuss a related problem.     

Vojta's Main Conjecture for blowup surfaces

In this paper, we prove Vojta's Main Conjecture for split blowups of products of certain elliptic curves with themselves. We then deduce from the conjecture bounds on the average number of rational points lying on curves on these surfaces, and expound upon this connection for abelian surfaces and rational surfaces.

     

Iwasawa theory of quadratic twists of <Emphasis Type="Italic">X</Emphasis><Subscript>0</Subscript>(49)

The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X ₀(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X ₀(49) by the quadratic extension \(KK(\sqrt M )/K\), where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F _∞ = K(E _p∞), where E _p∞ denotes the group of p^∞-division points on E. Moreover, writing B for the twist of X ₀(49) by \(K(\sqrt[4]{{ - 7}})/K\), our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z₂-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.     

Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

Property A and affine buildings

Yu's Property A is a non-equivariant generalisation of amenability introduced in his study of the coarse Baum Connes conjecture. In this paper we show that all affine buildings of type , and have Property A. Together with results of Guentner, Higson and Weinberger, this completes a programme to show that all affine building have Property A. In passing we use our technique to obtain a new proof for groups acting on buildings.