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arxiv: 2512.12841 · v2 · submitted 2025-12-14 · 🧮 math.NT · math.CO

Further Extensions of Sury's Identity

Gregory Dresden , Xiaoya Gao This is my paper

Pith reviewed 2026-05-16 22:17 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Sury's identityFibonacci numbersLucas numberssummation formulaspowers of twoelementary methodsrecurrence sequences
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The pith

A different elementary approach yields multiple new summation formulas extending Sury's identity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Sury's identity is a summation formula that links Lucas numbers, Fibonacci numbers, and powers of two. The paper develops an alternative method to produce several additional summation formulas of the same type. This method uses only elementary techniques yet reaches identities not previously recorded. A reader would care because the new formulas supply more closed-form relations among these sequences, expanding the practical tools available for summing them.

Core claim

By taking a different approach from prior extensions, the authors obtain a collection of new summation formulas that extend Sury's identity, all derived through direct elementary manipulations rather than more advanced machinery.

What carries the argument

The alternative elementary summation method that generates the new identities connecting Fibonacci and Lucas numbers to powers of two.

If this is right

  • The same method can be used to generate still more summation formulas for the same sequences.
  • The new identities supply additional closed forms for sums involving Fibonacci and Lucas numbers.
  • These formulas allow direct evaluation of certain infinite or finite sums without recursion.
  • The approach offers a systematic way to produce further extensions by varying the parameters in the sums.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may extend naturally to other linear recurrence sequences with similar generating functions.
  • Numerical verification for large indices would provide strong practical confirmation of the identities.
  • The identities could be re-derived via generating functions to reveal deeper algebraic structure.

Load-bearing premise

The new summation formulas are both valid and genuinely distinct from all previously published identities.

What would settle it

Substitute small integer values for the index into any one of the claimed new formulas and verify that the two sides are numerically equal.

read the original abstract

The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this paper, we take a different approach that allows us to produce a good number of new summation formulas, all from elementary (but non-trivial) methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends Sury's identity, a summation relating Lucas numbers, Fibonacci numbers, and powers of two, by deriving multiple new summation formulas via elementary methods based on recurrence relations and binomial expansions. Explicit derivations are given for new identities in theorems such as 2.1, 3.3, and 4.2, claimed to be algebraically distinct from prior results and obtained without generating functions.

Significance. If the derivations are correct, the work adds concrete new identities to the literature on linear recurrence sequences, which may support combinatorial interpretations or further extensions. The elementary approach is a strength, as it relies only on standard, reversible steps from established Fibonacci/Lucas properties and avoids heavier machinery.

minor comments (3)
  1. The abstract states that 'a good number of new summation formulas' are produced; specifying the exact count and briefly indicating their forms would improve clarity for readers.
  2. In the statements of Theorems 2.1, 3.3, and 4.2, the range of summation indices and any implicit assumptions on n should be stated explicitly at the outset of each theorem to avoid ambiguity.
  3. A short table or remark comparing the new identities to the original Sury formula and to the most closely related extensions in the cited literature would help readers assess novelty at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. We appreciate the recognition of the elementary approach and the new identities derived. As the report lists no specific major comments, we have used the opportunity to make minor editorial improvements to the presentation and clarity of the derivations in the revised version.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives multiple new summation identities extending Sury's formula using explicit, reversible steps based solely on standard Fibonacci/Lucas recurrence relations and binomial expansions (e.g., Theorems 2.1, 3.3, 4.2). No load-bearing self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work are present; all steps are algebraically independent of the target results and rely on externally established sequence properties.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard, externally known recurrence relations for Fibonacci and Lucas numbers with no free parameters or invented entities introduced.

axioms (1)
  • standard math Fibonacci and Lucas sequences obey the standard recurrences F_n = F_{n-1} + F_{n-2} and L_n = L_{n-1} + L_{n-2} with initial conditions F_0=0, F_1=1, L_0=2, L_1=1.
    Invoked implicitly to manipulate the summations in the new identities.

pith-pipeline@v0.9.0 · 5336 in / 1109 out tokens · 39878 ms · 2026-05-16T22:17:47.193082+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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