Admin: Create Project Euler/45 page with problem statement, approach, and solution link (via create-page on MediaWiki MCP Server)

2026-06-16T14:30:47Z

Create Project Euler/45 page with problem statement, approach, and solution link (via create-page on MediaWiki MCP Server)

New page

==Problem Statement==

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

{| class="wikitable"
|-
! Shape !! Formula !! First Terms
|-
| Triangle || <math>T_n = \frac{1}{2} n (n + 1)</math> || 1, 3, 6, 10, 15, ...
|-
| Pentagonal || <math>P_n = \frac{1}{2} n (3n - 1)</math> || 1, 5, 12, 22, 35, ...
|-
| Hexagonal || <math>H_n = n (2n - 1)</math> || 1, 6, 15, 28, 45, ...
|}

It can be verified that <math>T_{285} = P_{165} = H_{143} = 40755</math>.

Find the next triangle number that is also pentagonal and hexagonal.

==Approach==

===Key Insight: Hexagonal Numbers are Triangle Numbers===

Every hexagonal number is also a triangle number. Algebraically:

<math>H_n = n(2n - 1) = \frac{(2n-1)(2n)}{2} = T_{2n-1}</math>

This means the problem reduces to finding the next hexagonal number (after H<sub>143</sub> = 40755) that is also pentagonal. We only need to generate hexagonal numbers and test them for pentagonality.

===Testing for Pentagonal Numbers===

Given an integer ''x'', we can test whether it is pentagonal by solving the quadratic derived from the pentagonal formula:

<math>P_n = \frac{n(3n - 1)}{2} \implies 3n^2 - n - 2x = 0</math>

The discriminant is <math>1 + 24x</math>. If ''x'' is pentagonal, then:

* <math>s = \sqrt{1 + 24x}</math> must be an integer (a perfect square), ''and''
* <math>(s + 1) \bmod 6 = 0</math> (ensuring the root ''n'' is an integer).

Both conditions are checked in one pass for each candidate.

===Iteration Strategy===

# Start with <math>n = 144</math> (the next hexagonal index after 143, the known match).
# Compute the hexagonal number <math>H_n = n(2n - 1)</math>.
# Test whether <math>H_n</math> is pentagonal using the two conditions above.
# If it is pentagonal, this is the answer. Otherwise, increment ''n'' and repeat.

Because every hexagonal number is already triangle, no separate triangle check is needed.

==Solution==

Link: https://git.charlesreid1.com/cs/euler/src/branch/main/java/Problem045.java

==Flags==

{{ProjectEulerFlag}}

Project Euler/45 - Revision history

Admin: Create Project Euler/45 page with problem statement, approach, and solution link (via create-page on MediaWiki MCP Server)