[Paper Review] Mosaic supercongruences of Ramanujan-type

Jesús Guillera|arXiv (Cornell University)|Jul 14, 2010

Advanced Mathematical Identities3 references4 citations

TL;DR

This paper introduces 'mosaic supercongruences'—a generalization of Ramanujan-type supercongruences to hypergeometric series involving algebraic numbers, including square roots, in both the summands and the series parameters. It establishes that partial sums modulo $ p^3 $ exhibit congruences tied to Jacobi symbols and rational coefficients, extending Zudilin's and Van Hamme's patterns to multiquadratic fields.

ABSTRACT

We generalize the patterns of supercongruences of Ramanujan-type observed by L. Van Hamme and W. Zudilin to series involving simple square roots anywhere and not only in the result of the sum. To support our observations we give some examples.

Motivation & Objective

• To generalize supercongruence patterns observed by Van Hamme and Zudilin to Ramanujan-type series where algebraic numbers, including square roots, appear not only in the sum result but also in the summands.
• To establish a unified framework for supercongruences in hypergeometric series involving multiple square roots, such as $ \sqrt{d_1}, \dots, \sqrt{d_j} $, with rational coefficients.
• To conjecture and support analogous mosaic supercongruences for Ramanujan-like series involving $ 1/\pi^2 $, extending Zudilin's pattern to higher powers of $ p $.

Proposed method

• Define partial sums of Ramanujan-Sato-type series in the field $ \mathbb{Q}(\sqrt{d_1}, \dots, \sqrt{d_j}) $, expressing them as linear combinations of square roots with rational coefficients $ \alpha_i(p) $.
• Propose a new supercongruence pattern: for primes $ p > p_0 $, each coefficient $ \alpha_i(p) \equiv a_i \genfrac{(}{)}{0.8pt}{}{-d_i}{p} p \pmod{p^3} $, where $ a_i $ is the rational coefficient of $ \sqrt{d_i} $ in the constant term.
• Apply the framework to known Ramanujan-type series with complex algebraic parameters, including $ \sqrt{7}, \sqrt{-1}, \sqrt{5} $, and products like $ \sqrt{6} = \sqrt{2} \cdot \sqrt{3} $, to verify the pattern numerically.
• Use Zudilin’s translation method and modular forms to generate new series from known ones, preserving the supercongruence structure.
• Verify the congruences numerically for multiple examples, including Apéry, Domb, and Almkvist-Zudilin sequences, across various algebraic number fields.
• Generalize the pattern to $ 1/\pi^2 $-type series, where congruences hold modulo $ p^5 $, with coefficients matching the constant and linear terms in the series expansion.

Experimental results

Research questions

• RQ1 Can supercongruence patterns observed in Ramanujan-type series for $ 1/\pi $ be extended to cases where algebraic numbers, such as square roots, appear in the summands, not just in the sum result?
• RQ2 Do partial sums of such series, lying in multiquadratic fields $ \mathbb{Q}(\sqrt{d_1}, \dots, \sqrt{d_j}) $, satisfy congruences modulo $ p^3 $ involving Jacobi symbols and rational coefficients?
• RQ3 Can the Zudilin-type supercongruence pattern for $ 1/\pi^2 $-type series be generalized to include algebraic parameters, with congruences modulo $ p^5 $?
• RQ4 Is there a unifying algebraic structure—such as a Galois-invariant or modular form-based mechanism—that explains why these mosaic supercongruences hold across different number fields?
• RQ5 Can the method be extended to series with higher-degree algebraic numbers, such as $ \sqrt[3]{d} $, or more complex algebraic integers?

Key findings

• For the series involving $ \sqrt{15} $, the partial sum modulo $ p^3 $ satisfies $ \alpha_p \equiv 263 \genfrac{(}{)}{0.8pt}{}{-15}{p} p \pmod{p^3} $, matching Zudilin’s known result.
• In the $ \sqrt{7} $-based series, the coefficients $ \alpha_p \equiv -10 \genfrac{(}{)}{0.8pt}{}{-1}{p} p $ and $ \beta_p \equiv 7 \genfrac{(}{)}{0.8pt}{}{-7}{p} p \pmod{p^3} $, confirming the mosaic pattern.
• For the Apéry-based series with $ \sqrt{3} $ and $ \sqrt{15} $, the congruences $ \alpha_p \equiv -134 \genfrac{(}{)}{0.8pt}{}{-3}{p} p $ and $ \beta_p \equiv 60 \genfrac{(}{)}{0.8pt}{}{-15}{p} p \pmod{p^3} $ hold for $ p > 5 $.
• In the four-square-root series involving $ \sqrt{2}, \sqrt{3}, \sqrt{6} $, the coefficients satisfy $ \alpha_p \equiv 73 \genfrac{(}{)}{0.8pt}{}{-1}{p} p $, $ \beta_p \equiv 52 \genfrac{(}{)}{0.8pt}{}{-2}{p} p $, $ \gamma_p \equiv -42 \genfrac{(}{)}{0.8pt}{}{-3}{p} p $, and $ \delta_p \equiv -30 \genfrac{(}{)}{0.8pt}{}{-6}{p} p \pmod{p^3} $.
• For the complex series with $ \sqrt{-1} $ and $ \sqrt{7} $, the congruences $ \alpha_p \equiv -13p $ and $ \beta_p \equiv 7 \genfrac{(}{)}{0.8pt}{}{-7}{p} p \pmod{p^3} $ are verified numerically.
• For the $ 1/\pi^2 $-type series with $ \sqrt{5} $, the coefficients satisfy $ \alpha_p \equiv 56p^2 \pmod{p^5} $ and $ \beta_p \equiv -25 \genfrac{(}{)}{0.8pt}{}{5}{p} p^2 \pmod{p^5} $, generalizing Zudilin’s $ p^5 $-pattern.