Day 22 – Numerically 2026 Is Unremarkable Yet Happy

Introduction

This document (notebook) discusses number theory properties and relationships of the integer 2026.

The integer 2026 is semiprime and a happy number, with 365 as one of its primitive roots. Although 2026 may not be particularly noteworthy in number theory, this provides a great excuse to create various elaborate visualizations that reveal some interesting aspects of the number.

The computations in this document are done with the Raku package  “Math::NumberTheory”, [AAp1].

Setup

use Math::NumberTheory;
use Math::NumberTheory::Utilities;

use Data::Importers;
use Data::Translators;
use Data::TypeSystem;

use Graph;
use Graph::Classes;
use JavaScript::D3;

Notebook priming code:

#%javascript
require.config({
     paths: {
     d3: 'https://d3js.org/d3.v7.min'
}});

require(['d3'], function(d3) {
     console.log(d3);
});

2026 Is a Happy Semiprime with Primitive Roots

First, 2026 is obviously not prime—it is divisible by 2—but dividing it by 2 gives a prime, 1013:

is-prime(2026 / 2)

# True

Hence, 2026 is a semiprime. The integer 1013 is not a Gaussian prime, though:

is-prime(1013, :gaussian-integers)

# False

A happy number is a number for which iteratively summing the squares of its digits eventually reaches 1 (e.g., 13 → 10 → 1).
Here is a check that 2026 is happy:

is-happy-number(2026)

# True

Here is the corresponding trail of digit-square sums:

is-happy-number(2026, :trail).tail.head(*-1).join(' → ')

# 2026 → 44 → 32 → 13 → 10 → 1

Not many years in this century are happy numbers:

(2000...2100).grep({ is-happy-number($_) })

# (2003 2008 2019 2026 2030 2036 2039 2062 2063 2080 2091 2093)

The decomposition of 2026 as 2 * 1013 means the multiplicative group modulo 2026 has primitive roots. A primitive root exists for an integer n if and only if n is 1, 2, 4, p^k, or 2 p^k, where p is an odd prime and k > 0.

We can check additional facts about 2026, such as whether it is “square-free”, among other properties. However, let us compare these with the feature-rich 2025 in the next section.

Comparison with 2025

Here is a side-by-side comparison of key number theory properties for 2025 and 2026.

To summarize:

• 2025 stands out as a mathematically rich number, often highlighted in puzzles and articles for its perfect square status and connections to sums of cubes and odd numbers.
• 2026, in contrast, has fewer flashy properties — it’s a straightforward even semiprime — but it qualifies as a happy number and it has a primitive root.

Here is a computationally generated comparison table of most of the properties found in the table above:

#% html
my &divisors-count = { divisors($_).elems };
<is-prime is-composite divisors-count prime-omega euler-phi is-square-free is-happy-number is-harshad-number is-deficient-number primitive-root>.map({ %(sub => $_, '2025' => ::("&$_")(2025), '2026' => ::("&$_")(2026) ) })
==> to-html(field-names => ['sub', '2025', '2026'], align => 'left')

Phi Number System

Digits of 2026 represented in the Phi number system:

my @res = phi-number-system(2026);

# [15 13 10 6 -6 -11 -16]

Verification:

@res.map({ ϕ ** $_ }).sum.round(10e-11);

# 2026

Happy Numbers Trail Graph

Let us create and plot a graph showing the trails of all happy numbers less than or equal to 2026. Below, we identify these numbers and their corresponding trails:

my @ns = 1...2026;

my @trails = @ns.map({ is-happy-number($_):trail }).grep(*.head);

deduce-type(@trails)

# Vector((Any), 302)

Here is the corresponding trails graph, highlighting the primes and happy numbers:

#% html
my @prime-too = @trails.grep(*.head).map(*.tail.head).grep(*.&is-prime);
my @happy-too = @ns.grep(*.&is-harshad-number).grep(*.&is-happy-number);

my %highlight = '#006400' => @prime-too».Str,    # Deep Christmas green for primes
                'Blue' => [2026.Str, ],            # Blue for the special year
                '#fbb606ff' => @happy-too».Str;  # Darker gold for joyful numbers

my @edges = @trails.map({ $_.tail.head(*-1).rotor(2 => -1).map({ $_.head.Str => $_.tail.Str }) }).flat;

my $gTrails = Graph.new(@edges):!directed;

$gTrails.dot(
    engine => 'neato', 
    graph-size => 12, 
    vertex-shape => 'ellipse', vertex-height => 0.2, vertex-width => 0.4,
    :10vertex-font-size,
    vertex-color => '#B41E3A',
    vertex-fill-color => '#B41E3A',
    arrowsize => 0.6,
    edge-color => '#B41E3A',
    edge-width => 1.4,
    splines => 'curved',
    :%highlight
):svg

Triangular Numbers

There is a theorem by Gauss stating that any integer can be represented as a sum of at most three triangular numbers. Instead of programming such an integer decomposition representation in Raku, we can simply use Wolfram|Alpha, [AA1, AA3], or wolframscript to find an “interesting” solution:

#% bash
wolframscript -code 'FindInstance[{2026 == PolygonalNumber[i] + PolygonalNumber[j] + PolygonalNumber[k], i > 10, j > 10, k > 10}, {i, j, k}, Integers]'

# {{i -> 11, j -> 19, k -> 59}}

Here, we verify the result using Raku:

say "Triangular numbers : ", <11 19 59>.&polygonal-number(:3sides);
say "Sum : ", <11 19 59>.&polygonal-number(:3sides).sum;

# Triangular numbers : (66 190 1770)
# Sum : 2026

Chord Diagrams

Here is the number of primitive roots of the multiplication group modulo 2026:

primitive-root-list(2026).elems

# 440

Here are chord plots [AA2, AAp1, AAp2, AAv1] corresponding to a few selected primitive roots:

#% js
my $n = 2026;
<339 365 1529>.map( -> $p { 
    my &f = -> $x { power-mod($p, $x, $n) => power-mod($p, $x + 1, $n) };
    
    my @segments = circular-chords-tracing($n, with => &f, :d);
    
    @segments .= map({ $_<group> = $_<group>.Str; $_ });
    
    js-d3-list-line-plot(
        @segments,
        stroke-width => 0.1,
        background => '#1F1F1F',
        :300width, :300height,
        :!axes,
        :!legends,
        :10margins,
        color-scheme => 'Ivory',
        #:$title-color, title => $p
        )
}).join("\n")

Remark: It is interesting that 365 (the number of days in a common calendar year) is a primitive root of the multiplicative group generated by 2026. Not many years have this property this century; many do not have primitive roots at all.

(2000...2100).hyper(:4degree).grep({ 365 ∈ primitive-root-list($_) })

# (2003 2026 2039 2053 2063 2078 2089)

Quartic Graphs

The number 2026 appears in 18 results of the search “2026 graphs” in «The On-line Encyclopedia of Integer Sequences». Here is the first result (from 2025-12-17): A033483, “Number of disconnected 4-valent (or quartic) graphs with n nodes.” Below, we ingest the internal format of A033483’s page:

my $internal = data-import('https://oeis.org/A033483/internal', 'plaintext');
text-stats($internal)

# (chars => 2928 words => 383 lines => 98)

Here, we verify the title:

with $internal.match(/ '%' N (<-[%]>*)? <?before '%'> /) { $0.Str.trim }

# Number of disconnected 4-valent (or quartic) graphs with n nodes.

Here, we get the corresponding sequence:

my @seq = do with data-import('https://oeis.org/A033483/list', 'plaintext').match(/'[' (.*) ']'/) {
    $0.Str.split(',')».trim
}

# [0 0 0 0 0 0 0 0 0 0 1 1 3 8 25 88 378 2026 13351 104595 930586 9124662 96699987 1095469608 13175272208 167460699184 2241578965849 31510542635443 464047929509794 7143991172244290 114749135506381940 1919658575933845129 33393712487076999918 603152722419661386031]

Here we find the position of 2026 in that sequence:

@seq.grep(2026):k

# (17)

Given the title of the sequence and the extracted position of 2026, this means that the number of disconnected 4-regular graphs with 17 vertices is 2026. Let us create a few graphs from that set by using the 5-vertex complete graph (K₅) and circulant graphs.
Here is an example of such a graph:

#% html
reduce(    
    { $^a.union($^b) },
    [
        Graph::Complete.new(5),
        Graph::Complete.new(5).index-graph(6),
        Graph::Circulant.new(7,[1,2]).index-graph(11)
    ]
).dot(engine => 'neato'):svg

And here is another one:

#% html
my $g = reduce(    
    { $^a.union($^b) },
    [
        Graph::Complete.new(5).index-graph(13),
        Graph::Circulant.new(12, [1, 5]).index-graph(1),
    ]
);
$g.dot(engine => 'neato', splines => 'curved'):svg

Here, we check that all vertices have degree 4:

$g.vertex-degree.sum / $g.vertex-count

# 4

Remark: Note that although the plots show disjoint graphs, each graph plot represents a single graph object.

Ways to compute 2026

Here are a few ways to compute 2026:

sub postfix:<!>(UInt:D $n) { [*] 1..$n }

[ 
    2**11 - 22,
    1 + 2 * 3 / 4 / 5 / 6 * 7! * 8 + 9,
    1 + 2 + 3 * 4! + 5 / 6! * 7 * 8! - 9,
    1 + 2 + 345 * 6 - 7 * 8 + 9,
    1 - 2 * 3! * 456 * 7 + 8! + 9,
    12 / 3! * 4**5 + 67 - 89,
    12**3 + 45 / 6! * 7! - 8 - 9,
    123 / 4! * 56 * 7 + 8 + 9,
    9! / 8 / 7 / 6 + 5**4 + 321,  
    9 * 8 - 7 + 654 * 3 - 2 + 1,
    987 + 6! + 5 * 4**3 - 2 + 1
]

# [2026 2026 2026 2026 2026 2026 2026 2026 2026 2026 2026]

Additional Comments

This section has a few additional (leftover) comments.

• After I researched and published the blog post “Numeric properties of 2025”, [AA1], in the first few days of 2025, I decided to program additional Number theory functionalities for Raku — see the package “Math::NumberTheory”, [AAp1].
• Most notably, “Math::NumberTheory” is extended to work with Gaussian integers, and the operators GCD and LCD are also extended to work with rationals.
• One of my current goals is to make Raku extremely good for researching Number theory phenomena.
  • One of the most significant features of Raku is its bignum arithmetic and its good built-in numeric functions applicable in Number theory.
• Number theory provides many opportunities for visualizations, so I included utilities for some of the popular patterns in “Math::NumberTheory”, [AAp1, AAp2].
  • I use the package “JavaScript::D3”, [AAp3], for almost all Number theory visualizations with Raku.
  • Often, I visualize associated graphs using the DOT language specs provided by the package “Graph”, [AAp4] (as seen above).
    • The Graphviz engines generate images in different formats using DOT specs.
• The number of years in this century that have primitive roots and have 365 as a primitive root is less than the number of years that are happy numbers.
• I would say I spent too much time finding a good, suitable, Christmas-themed combination of colors for the trails graph.
• To get the quartic graph counting sequence A033483, I tried to use “Math::Sequences”, but since that package does not provide the sequence, I used an ad hoc retrieval (for which Raku is perfect.)
• While working on this document, I implemented in “Math::NumberTheory” a set of new functions: integer-digits, is-happy-number, is-harshad-number, is-abundant-number, is-perfect-number, is-deficient-number, abundant-number, deficient-number, and perfect-number.
  • All of these functions — except integer-digits — had lower implementation priority.
• Another implementation effort was to finally come up with a Command Line Interface (CLI).
  • I advocate that a CLI should be considered for all Raku packages, and most should have one.
    
    

#%bash
number-theory is happy number 2026

# True

#%bash
number-theory primitive root list 2026 | grep 365

# 365

References

Articles, blog posts

[AA1] Anton Antonov, “Numeric properties of 2025”, (2025), RakuForPrediction at WordPress.

[AA2] Anton Antonov, “Primitive roots generation trails”, (2025), MathematicaForPrediction at WordPress.

[AA3] Anton Antonov, “Chatbook New Magic Cells”, (2024), RakuForPrediction at WordPress.

[EW1] Eric W. Weisstein, “Quartic Graph”. From MathWorld–A Wolfram Resource.

Notebooks

[AAn1] Anton Antonov, “Primitive roots generation trails”, (2025), Wolfram Community. STAFFPICKS, April 9, 2025​.

[EPn1] Ed Pegg, “Happy 2025 =1³+2³+3³+4³+5³+6³+7³+8³+9³!”, ​Wolfram Community, STAFFPICKS, December 30, 2024​.

Packages, paclets

[AAp1] Anton Antonov, Math::NumberTheory, Raku package, (2025), GitHub/antononcube.

[AAp2] Anton Antonov,
NumberTheoryUtilities, Wolfram Language paclet, (2025), Wolfram Language Paclet Repository.

[AAp3] Anton Antonov, JavaScript::D3, Raku package, (2021-2025), GitHub/antononcube.

[AAp4] Anton Antonov, Graph, Raku package, (2024-2025), GitHub/antononcube.

Videos

[AAv1] Anton Antonov, Number theory neat examples in Raku (Set 3), (2025), YouTube/@AAA4prediction.