Square Root of Natural Numbers as an infinite Staircase of Continued Fractions Article Swipe

View

Bhatnagar Anuj ·

YOU? · · 2026 · Open Access · · DOI: https://doi.org/10.5281/zenodo.17699284

When one lists the continued fractions of √ n across the natural numbers, a striking regularity appears: the coefficients rise in discrete steps, forming an arithmetical and visual staircase. This article reveals the algebraic and geometric underpinnings of that pattern. Each segment between consecutive perfect squares yields a constant pair of leading coefficients and a linearly increasing remainder, giving rise to a simple integer structure that unites the periodicity of quadratic irrationals with an illuminating global picture. We include a short proof of the fundamental relation and three complementary visualizations, along with a brief note on the golden ratio as a conceptual ”zeroth stair.

Related Topics

Mathematical Analysis

Arithmetic

Concepts

Mathematics Natural number Algebraic number Simple (philosophy) Arithmetic function Square root Integer (computer science) Combinatorics Constant (computer programming) Square (algebra) Number theory Quadratic equation Square number Discrete mathematics Pure mathematics Relation (database) Algebraic geometry Fraction (chemistry) Square-free integer Mathematical analysis Multiple Arithmetic Golden ratio Geometric progression Natural (archaeology)

Metadata

Type: preprint
Language: en
Landing Page: https://doi.org/10.5281/zenodo.17699284
OA Status: green
OpenAlex ID: https://openalex.org/W7106500450

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W7106500450

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.5281/zenodo.17699284

Digital Object Identifier
Title: Square Root of Natural Numbers as an infinite Staircase of Continued Fractions

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2026

Year of publication
Publication date: 2026-11-24

Full publication date if available
Authors: Bhatnagar Anuj

List of authors in order
Landing page: https://doi.org/10.5281/zenodo.17699284

Publisher landing page
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://doi.org/10.5281/zenodo.17699284

Direct OA link when available
Concepts: Mathematics, Natural number, Algebraic number, Simple (philosophy), Arithmetic function, Square root, Integer (computer science), Combinatorics, Constant (computer programming), Square (algebra), Number theory, Quadratic equation, Square number, Discrete mathematics, Pure mathematics, Relation (database), Algebraic geometry, Fraction (chemistry), Square-free integer, Mathematical analysis, Multiple, Arithmetic, Golden ratio, Geometric progression, Natural (archaeology)

Top concepts (fields/topics) attached by OpenAlex
Cited by: 0

Total citation count in OpenAlex

Full payload

id	https://openalex.org/W7106500450
doi	https://doi.org/10.5281/zenodo.17699284
ids.doi	https://doi.org/10.5281/zenodo.17699284
ids.openalex	https://openalex.org/W7106500450
fwci	0.0
type	preprint
title	Square Root of Natural Numbers as an infinite Staircase of Continued Fractions
biblio.issue
biblio.volume
biblio.last_page
biblio.first_page
is_xpac	False
apc_list
apc_paid
concepts[0].id	https://openalex.org/C33923547
concepts[0].level	0
concepts[0].score	0.8778542280197144
concepts[0].wikidata	https://www.wikidata.org/wiki/Q395
concepts[0].display_name	Mathematics
concepts[1].id	https://openalex.org/C184264201
concepts[1].level	2
concepts[1].score	0.6315406560897827
concepts[1].wikidata	https://www.wikidata.org/wiki/Q21199
concepts[1].display_name	Natural number
concepts[2].id	https://openalex.org/C9376300
concepts[2].level	2
concepts[2].score	0.6187198758125305
concepts[2].wikidata	https://www.wikidata.org/wiki/Q168817
concepts[2].display_name	Algebraic number
concepts[3].id	https://openalex.org/C2780586882
concepts[3].level	2
concepts[3].score	0.5901679992675781
concepts[3].wikidata	https://www.wikidata.org/wiki/Q7520643
concepts[3].display_name	Simple (philosophy)
concepts[4].id	https://openalex.org/C112343008
concepts[4].level	2
concepts[4].score	0.5869834423065186
concepts[4].wikidata	https://www.wikidata.org/wiki/Q140744
concepts[4].display_name	Arithmetic function
concepts[5].id	https://openalex.org/C11577676
concepts[5].level	2
concepts[5].score	0.5869619846343994
concepts[5].wikidata	https://www.wikidata.org/wiki/Q134237
concepts[5].display_name	Square root
concepts[6].id	https://openalex.org/C97137487
concepts[6].level	2
concepts[6].score	0.5774396061897278
concepts[6].wikidata	https://www.wikidata.org/wiki/Q729138
concepts[6].display_name	Integer (computer science)
concepts[7].id	https://openalex.org/C114614502
concepts[7].level	1
concepts[7].score	0.5486741662025452
concepts[7].wikidata	https://www.wikidata.org/wiki/Q76592
concepts[7].display_name	Combinatorics
concepts[8].id	https://openalex.org/C2777027219
concepts[8].level	2
concepts[8].score	0.546109676361084
concepts[8].wikidata	https://www.wikidata.org/wiki/Q1284190
concepts[8].display_name	Constant (computer programming)
concepts[9].id	https://openalex.org/C135692309
concepts[9].level	2
concepts[9].score	0.5394142270088196
concepts[9].wikidata	https://www.wikidata.org/wiki/Q111124
concepts[9].display_name	Square (algebra)
concepts[10].id	https://openalex.org/C169654258
concepts[10].level	2
concepts[10].score	0.4520373046398163
concepts[10].wikidata	https://www.wikidata.org/wiki/Q12479
concepts[10].display_name	Number theory
concepts[11].id	https://openalex.org/C129844170
concepts[11].level	2
concepts[11].score	0.4434051811695099
concepts[11].wikidata	https://www.wikidata.org/wiki/Q41299
concepts[11].display_name	Quadratic equation
concepts[12].id	https://openalex.org/C173808221
concepts[12].level	2
concepts[12].score	0.40764737129211426
concepts[12].wikidata	https://www.wikidata.org/wiki/Q50705
concepts[12].display_name	Square number
concepts[13].id	https://openalex.org/C118615104
concepts[13].level	1
concepts[13].score	0.38452020287513733
concepts[13].wikidata	https://www.wikidata.org/wiki/Q121416
concepts[13].display_name	Discrete mathematics
concepts[14].id	https://openalex.org/C202444582
concepts[14].level	1
concepts[14].score	0.3412432372570038
concepts[14].wikidata	https://www.wikidata.org/wiki/Q837863
concepts[14].display_name	Pure mathematics
concepts[15].id	https://openalex.org/C25343380
concepts[15].level	2
concepts[15].score	0.32906633615493774
concepts[15].wikidata	https://www.wikidata.org/wiki/Q277521
concepts[15].display_name	Relation (database)
concepts[16].id	https://openalex.org/C68363185
concepts[16].level	2
concepts[16].score	0.3270401358604431
concepts[16].wikidata	https://www.wikidata.org/wiki/Q180969
concepts[16].display_name	Algebraic geometry
concepts[17].id	https://openalex.org/C149629883
concepts[17].level	2
concepts[17].score	0.3254373371601105
concepts[17].wikidata	https://www.wikidata.org/wiki/Q660926
concepts[17].display_name	Fraction (chemistry)
concepts[18].id	https://openalex.org/C188139237
concepts[18].level	2
concepts[18].score	0.314749538898468
concepts[18].wikidata	https://www.wikidata.org/wiki/Q50706
concepts[18].display_name	Square-free integer
concepts[19].id	https://openalex.org/C134306372
concepts[19].level	1
concepts[19].score	0.2990979552268982
concepts[19].wikidata	https://www.wikidata.org/wiki/Q7754
concepts[19].display_name	Mathematical analysis
concepts[20].id	https://openalex.org/C164680029
concepts[20].level	2
concepts[20].score	0.28126809000968933
concepts[20].wikidata	https://www.wikidata.org/wiki/Q1055293
concepts[20].display_name	Multiple
concepts[21].id	https://openalex.org/C94375191
concepts[21].level	1
concepts[21].score	0.27091899514198303
concepts[21].wikidata	https://www.wikidata.org/wiki/Q11205
concepts[21].display_name	Arithmetic
concepts[22].id	https://openalex.org/C18802822
concepts[22].level	2
concepts[22].score	0.2677929103374481
concepts[22].wikidata	https://www.wikidata.org/wiki/Q41690
concepts[22].display_name	Golden ratio
concepts[23].id	https://openalex.org/C14367171
concepts[23].level	2
concepts[23].score	0.2590814232826233
concepts[23].wikidata	https://www.wikidata.org/wiki/Q173523
concepts[23].display_name	Geometric progression
concepts[24].id	https://openalex.org/C2776608160
concepts[24].level	2
concepts[24].score	0.2552325427532196
concepts[24].wikidata	https://www.wikidata.org/wiki/Q4785462
concepts[24].display_name	Natural (archaeology)
keywords[0].id	https://openalex.org/keywords/natural-number
keywords[0].score	0.6315406560897827
keywords[0].display_name	Natural number
keywords[1].id	https://openalex.org/keywords/algebraic-number
keywords[1].score	0.6187198758125305
keywords[1].display_name	Algebraic number
keywords[2].id	https://openalex.org/keywords/simple
keywords[2].score	0.5901679992675781
keywords[2].display_name	Simple (philosophy)
keywords[3].id	https://openalex.org/keywords/arithmetic-function
keywords[3].score	0.5869834423065186
keywords[3].display_name	Arithmetic function
keywords[4].id	https://openalex.org/keywords/square-root
keywords[4].score	0.5869619846343994
keywords[4].display_name	Square root
keywords[5].id	https://openalex.org/keywords/integer
keywords[5].score	0.5774396061897278
keywords[5].display_name	Integer (computer science)
keywords[6].id	https://openalex.org/keywords/constant
keywords[6].score	0.546109676361084
keywords[6].display_name	Constant (computer programming)
keywords[7].id	https://openalex.org/keywords/square
keywords[7].score	0.5394142270088196
keywords[7].display_name	Square (algebra)
language	en
locations[0].id	doi:10.5281/zenodo.17699284
locations[0].is_oa	True
locations[0].source.id	https://openalex.org/S4306400562
locations[0].source.issn
locations[0].source.type	repository
locations[0].source.is_oa	True
locations[0].source.issn_l
locations[0].source.is_core	False
locations[0].source.is_in_doaj	False
locations[0].source.display_name	Zenodo (CERN European Organization for Nuclear Research)
locations[0].source.host_organization	https://openalex.org/I67311998
locations[0].source.host_organization_name	European Organization for Nuclear Research
locations[0].source.host_organization_lineage	https://openalex.org/I67311998
locations[0].license	cc-by
locations[0].pdf_url
locations[0].version
locations[0].raw_type	article
locations[0].license_id	https://openalex.org/licenses/cc-by
locations[0].is_accepted	False
locations[0].is_published
locations[0].raw_source_name
locations[0].landing_page_url	https://doi.org/10.5281/zenodo.17699284
indexed_in	datacite
authorships[0].author.id	https://openalex.org/A2748699318
authorships[0].author.orcid
authorships[0].author.display_name	Bhatnagar Anuj
authorships[0].countries	IN
authorships[0].affiliations[0].institution_ids	https://openalex.org/I4210137186
authorships[0].affiliations[0].raw_affiliation_string	Society for Applied Microwave Electronics Engineering & Research
authorships[0].institutions[0].id	https://openalex.org/I4210137186
authorships[0].institutions[0].ror	https://ror.org/https://ror.org/03wa5d325
authorships[0].institutions[0].type	facility
authorships[0].institutions[0].lineage	https://openalex.org/I4210137186
authorships[0].institutions[0].country_code	IN
authorships[0].institutions[0].display_name	Society for Applied Microwave Electronics Engineering & Research
authorships[0].author_position	first
authorships[0].raw_author_name	bhatnagar, anuj
authorships[0].is_corresponding	True
authorships[0].raw_affiliation_strings	Society for Applied Microwave Electronics Engineering & Research
has_content.pdf	False
has_content.grobid_xml	False
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://doi.org/10.5281/zenodo.17699284
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2025-11-25T00:00:00
display_name	Square Root of Natural Numbers as an infinite Staircase of Continued Fractions
has_fulltext	False
is_retracted	False
updated_date	2025-11-25T14:47:58.456640
primary_topic
cited_by_count	0
locations_count	1
best_oa_location.id	doi:10.5281/zenodo.17699284
best_oa_location.is_oa	True
best_oa_location.source.id	https://openalex.org/S4306400562
best_oa_location.source.issn
best_oa_location.source.type	repository
best_oa_location.source.is_oa	True
best_oa_location.source.issn_l
best_oa_location.source.is_core	False
best_oa_location.source.is_in_doaj	False
best_oa_location.source.display_name	Zenodo (CERN European Organization for Nuclear Research)
best_oa_location.source.host_organization	https://openalex.org/I67311998
best_oa_location.source.host_organization_name	European Organization for Nuclear Research
best_oa_location.source.host_organization_lineage	https://openalex.org/I67311998
best_oa_location.license	cc-by
best_oa_location.pdf_url
best_oa_location.version
best_oa_location.raw_type	article
best_oa_location.license_id	https://openalex.org/licenses/cc-by
best_oa_location.is_accepted	False
best_oa_location.is_published	False
best_oa_location.raw_source_name
best_oa_location.landing_page_url	https://doi.org/10.5281/zenodo.17699284
primary_location.id	doi:10.5281/zenodo.17699284
primary_location.is_oa	True
primary_location.source.id	https://openalex.org/S4306400562
primary_location.source.issn
primary_location.source.type	repository
primary_location.source.is_oa	True
primary_location.source.issn_l
primary_location.source.is_core	False
primary_location.source.is_in_doaj	False
primary_location.source.display_name	Zenodo (CERN European Organization for Nuclear Research)
primary_location.source.host_organization	https://openalex.org/I67311998
primary_location.source.host_organization_name	European Organization for Nuclear Research
primary_location.source.host_organization_lineage	https://openalex.org/I67311998
primary_location.license	cc-by
primary_location.pdf_url
primary_location.version
primary_location.raw_type	article
primary_location.license_id	https://openalex.org/licenses/cc-by
primary_location.is_accepted	False
primary_location.is_published	False
primary_location.raw_source_name
primary_location.landing_page_url	https://doi.org/10.5281/zenodo.17699284
publication_date	2026-11-24
publication_year	2026
referenced_works_count	0
abstract_inverted_index.a	13, 47, 54, 61, 79, 92, 100
abstract_inverted_index.n	8
abstract_inverted_index.We	77
abstract_inverted_index.an	24, 73
abstract_inverted_index.as	99
abstract_inverted_index.in	20
abstract_inverted_index.of	6, 37, 50, 69, 82
abstract_inverted_index.on	95
abstract_inverted_index.to	60
abstract_inverted_index.and	26, 34, 53, 86
abstract_inverted_index.one	1
abstract_inverted_index.the	3, 10, 17, 32, 67, 83, 96
abstract_inverted_index.√	7
abstract_inverted_index.Each	40
abstract_inverted_index.This	29
abstract_inverted_index.When	0
abstract_inverted_index.note	94
abstract_inverted_index.pair	49
abstract_inverted_index.rise	19, 59
abstract_inverted_index.that	38, 65
abstract_inverted_index.with	72, 91
abstract_inverted_index.along	90
abstract_inverted_index.brief	93
abstract_inverted_index.lists	2
abstract_inverted_index.proof	81
abstract_inverted_index.ratio	98
abstract_inverted_index.short	80
abstract_inverted_index.three	87
abstract_inverted_index.across	9
abstract_inverted_index.giving	58
abstract_inverted_index.global	75
abstract_inverted_index.golden	97
abstract_inverted_index.simple	62
abstract_inverted_index.stair.	103
abstract_inverted_index.steps,	22
abstract_inverted_index.unites	66
abstract_inverted_index.visual	27
abstract_inverted_index.yields	46
abstract_inverted_index.article	30
abstract_inverted_index.between	42
abstract_inverted_index.forming	23
abstract_inverted_index.include	78
abstract_inverted_index.integer	63
abstract_inverted_index.leading	51
abstract_inverted_index.natural	11
abstract_inverted_index.perfect	44
abstract_inverted_index.reveals	31
abstract_inverted_index.segment	41
abstract_inverted_index.squares	45
abstract_inverted_index.appears:	16
abstract_inverted_index.constant	48
abstract_inverted_index.discrete	21
abstract_inverted_index.linearly	55
abstract_inverted_index.numbers,	12
abstract_inverted_index.pattern.	39
abstract_inverted_index.picture.	76
abstract_inverted_index.relation	85
abstract_inverted_index.striking	14
abstract_inverted_index.algebraic	33
abstract_inverted_index.continued	4
abstract_inverted_index.fractions	5
abstract_inverted_index.geometric	35
abstract_inverted_index.quadratic	70
abstract_inverted_index.structure	64
abstract_inverted_index.”zeroth	102
abstract_inverted_index.conceptual	101
abstract_inverted_index.increasing	56
abstract_inverted_index.regularity	15
abstract_inverted_index.remainder,	57
abstract_inverted_index.staircase.	28
abstract_inverted_index.consecutive	43
abstract_inverted_index.fundamental	84
abstract_inverted_index.irrationals	71
abstract_inverted_index.periodicity	68
abstract_inverted_index.arithmetical	25
abstract_inverted_index.coefficients	18, 52
abstract_inverted_index.illuminating	74
abstract_inverted_index.complementary	88
abstract_inverted_index.underpinnings	36
abstract_inverted_index.visualizations,	89
cited_by_percentile_year
countries_distinct_count	1
institutions_distinct_count	1
citation_normalized_percentile