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Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Zhi-Hong Sun (2011)

Acta Arithmetica

Congruences for certain families of Apéry-like sequences

Zhi-Hong Sun (2022)

Czechoslovak Mathematical Journal

We systematically investigate the expressions and congruences for both a one-parameter family ${G_{n} (x)}$ as well as a two-parameter family ${G_{n} (r, m)}$ of sequences.

Congruences for $q^{[p / 8]} (m o d p)$

Zhi-Hong Sun (2013)

Acta Arithmetica

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p / 8]} (m o d p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.

Displaying 1 – 6 of 6

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Congruences for certain families of Apéry-like sequences

Congruences for $q^{[p / 8]} (m o d p)$

Construction of entire modular forms of weights 5 and 6 for the congruence group $Γ_{0} (4 N)$ .

Correlations of representation functions of binary quadratic forms

Cyclic polygons of integer points

Currently displaying 1 – 6 of 6

Displaying 1 – 6 of 6

Congruences for ( A + √ ( A ² + m B ² ) ) ( p - 1 ) / 2 ) and ( b + √ ( a ² + b ² ) ) ( p - 1 ) / 4 ( m o d p )

Congruences for certain families of Apéry-like sequences

Congruences for q [ p / 8 ] ( m o d p )

Construction of entire modular forms of weights 5 and 6 for the congruence group Γ 0 ( 4 N ) .

Correlations of representation functions of binary quadratic forms

Cyclic polygons of integer points

Currently displaying 1 – 6 of 6

Congruences for ${(A + \sqrt (A ² + m B ²))}^{(p - 1) / 2)}$ and ${(b + \sqrt (a ² + b ²))}^{(p - 1) / 4} (m o d p)$

Congruences for $q^{[p / 8]} (m o d p)$

Construction of entire modular forms of weights 5 and 6 for the congruence group $Γ_{0} (4 N)$ .