Introduction

This document 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.

Setup

In[]:=
(*PacletInstall["AntonAntonov/NumberTheoryUtilities"]*)​​Needs["AntonAntonov`NumberTheoryUtilities`"]

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
:
In[]:=
PrimeQ[2026/2]
Out[]=
True
Hence, 2026 is a semiprime . The integer 1013 is not a Gaussian prime , though:
In[]:=
PrimeQ[1013,GaussianIntegers->True]
Out[]=
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:
In[]:=
ResourceFunction["HappyNumberQ"][2026]
Out[]=
True
Here is the corresponding trail of digit-square sums:
In[]:=
FixedPointList[Total[IntegerDigits[#]^2]&,2026]
Out[]=
{2026,44,32,13,10,1,1}
Not many years in this century are happy numbers:
In[]:=
Pick[Range[2000,2100],ResourceFunction["HappyNumberQ"]/@Range[2000,2100]]
Out[]=
{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
,
k
p
, or
2
k
p
, where
k
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
.
Out[]=
#
Property
2025
2026
Notes
1
Prime or Composite
Composite
Composite
Both non-prime.
2
Prime Factorization
3^4 × 5^2 (81 × 25)
2 × 1013
2025 has repeated small primes; 2026 is a semiprime (product of two distinct primes).
3
Number of Divisors
15 (highly composite for its size)
4 (1, 2, 1013, 2026)
2025 has many divisors; 2026 has very few.
4
Perfect Square
Yes (45^2 = 2025)
No
Major highlight for 2025—rare square year.
5
Sum of Cubes
Yes (1^3 + 2^3 + ... + 9^3 = (1 + ... + 9)^2 = 2025)
No
Iconic property for 2025 (Nicomachus's theorem).
6
Happy Number
No (process leads to cycle including 4)
Yes (repeated squared digit sums reach 1)
Key point for 2026—its main "happy" trait.
7
Harshad Number
Yes (divisible by 9)
No (not divisible by 10)
2025 qualifies; 2026 does not.
8
Primitive Roots
No
Yes
This is a relatively rare property to have.
9
Other Notable Traits
{(20 + 25)^2 = 2025,Sum of first 45 odd numbers,Deficient number,Many pattern-based representations}
{Even number,Deficient number,Few special patterns}
2025 is packed with elegant properties; 2026 is more "plain" beyond being happy.
10
Overall "Interest" Level
Highly interesting—celebrated in math communities for squares, cubes, and patterns
Relatively uninteresting—basic semiprime with no standout geometric or sum properties
Reinforces blog's angle.
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:
    In[]:=
    Dataset@Map[<|"Function"->#1,"2025"->#1[2025],"2026"->#1[2026]|>&,{PrimeQ,CompositeQ,Length@*Divisors,PrimeOmega,EulerPhi,SquareFreeQ,ResourceFunction["HappyNumberQ"],ResourceFunction["HarshadNumberQ"],ResourceFunction["DeficientNumberQ"],PrimitiveRoot}]
    Out[]=
    Function
    2025
    2026
    PrimeQ
    False
    False
    CompositeQ
    True
    True
    Length@*Divisors
    15
    4
    PrimeOmega
    6
    2
    EulerPhi
    1080
    1012
    SquareFreeQ
    False
    True
    [◼]
    HappyNumberQ
    False
    True
    [◼]
    HarshadNumberQ
    True
    False
    [◼]
    DeficientNumberQ
    True
    True
    PrimitiveRoot
    PrimitiveRoot[2025]
    3

    Phi Number System

    Digits of 2026 represented in the Phi number system :
    In[]:=
    ResourceFunction["PhiNumberSystem"][2026]
    Out[]=
    {15,13,10,6,-6,-11,-16}
    Verification:
    In[]:=
    Total[GoldenRatio^%]//RootReduce
    Out[]=
    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:
    In[]:=
    ns=Range[2,2026];​​AbsoluteTiming[​​trails=Map[FixedPointList[Total[IntegerDigits[#]^2]&,#,100,SameTest(Abs[#1-#2]<1*^-10&)]&,ns];​​]
    Out[]=
    {0.29022,Null}
    Here is the corresponding trails graph, highlighting the primes and happy numbers:
    In[]:=
    happy=First/@Select[trails,#〚-1〛==1&];​​primeToo=Select[happy,PrimeQ];​​joyfulToo=Select[happy,ResourceFunction["HarshadNumberQ"]];​​aColors=Flatten@{Thread[primeToo->ResourceFunction["HexToColor"]["#006400"]],2026->Blue,Thread[joyfulToo->ResourceFunction["HexToColor"]["#fbb606ff"]],_->ResourceFunction["HexToColor"]["#B41E3A"]};​​edges=DeleteDuplicates@Flatten@Map[Rule@@@Partition[Most[#],2,1]&,Select[trails,#〚-1〛==1&]];​​vf1[{xc_,yc_},name_,{w_,h_}]:={(name/.aColors),EdgeForm[name/.aColors],Rectangle[{xc-2w,yc-h},{xc+2w,yc+h}],Text[Style[name,12,White],{xc,yc}]}​​vf2[{xc_,yc_},name_,{w_,h_}]:={(name/.aColors),EdgeForm[name/.aColors],Disk[{xc,yc},{2w,h}],Text[Style[name,12,White],{xc,yc}]}​​​​gTrails=​​Graph[​​edges,​​VertexStyle->ResourceFunction["HexToColor"]["#B41E3A"],VertexSize->1.8,​​VertexShapeFunctionvf2,​​EdgeStyle->Directive[ResourceFunction["HexToColor"]["#B41E3A"]],​​EdgeShapeFunction({ResourceFunction["HexToColor"]["#B41E3A"],Thick,BezierCurve[#1]}&),​​DirectedEdges->False,​​GraphLayout->"SpringEmbedding",​​ImageSize->1000]
    Out[]=

    Triangular Numbers

    There is a theorem by Gauss stating that any integer can be represented as a sum of at most three triangular numbers. Here we find an "interesting" solution:
    Here, we verify the result:

    Chord Diagrams

    Here is the number of primitive roots of the multiplication group modulo 2026:
    Here are chord plots [AA2, AAp1, AAp2, AAv1] corresponding to a few selected primitive roots:
    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.

    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 retrieve properties from A033483's page:
    Here, we just get the title:
    Here, we get the corresponding sequence:
    Here we find the position of 2026 in that sequence:
    And here is another one:
    Here, we check that all vertices have degree 4:
    Remark: Note that although the plots show disjoint graphs, each graph plot represents a single graph object.

    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].
    • ◼
  • That Raku blog post was inspired by Ed Pegg's Wolfram Community post, [EPn1].
    • ◼
  • Number theory provides many opportunities for visualizations, so I included utilities for some of the popular patterns in "Math::NumberTheory", [AAp1] and "NumberTheoryUtilities".
    • ◼
  • 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.

    • 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​.

    • Functions, 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 .
    • ◼
  • [JFf1] Jesse Friedman, OEISSequenceData, (2019-2024), Wolfram Function Repository.
    • ◼
  • [MSf1] Michael Solami, HexToColor, (2020), Wolfram Function Repository.
    • ◼
  • [SHf1] Sander Huisman, HappyNumberQ, (2019), Wolfram Function Repository.
    • ◼
  • [SHf2] Sander Huisman, HarshadNumberQ, (2023), Wolfram Function Repository.
    • ◼
  • [WAf1] Wolfram|Alpha Math Team, DeficientNumberQ, (2020-2023), Wolfram Function Repository.

    • Videos

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

    • CITE THIS NOTEBOOK

    Numerically 2026 is unremarkable yet happy: semiprime with primitive roots​
    by Anton Antonov​
    Wolfram Community, STAFF PICKS, December 22, 2025
    ​https://community.wolfram.com/groups/-/m/t/3594686