README.md

Benchmarked Performance Gains across Input Sizes (k)

The logs below capture how the optimized generation algorithm fundamentally eliminates the performance walls found in the original brute-force sandbox.

By replacing sequential integer factorization checks with an $O(k)$ direct-generation generator stream, computing $k = 1000$ squares no longer takes 2.25 hours (8098.075s), it completes instantly in 0.363 seconds. Even when scaling the input workload exponentially to find 1,000,000 perfect squares, the total execution time stays well under six seconds.

Detailed computation logs (tracking performance maps by loop iteration, discrete square number discovery intervals, and total execution cycles) are cataloged below:

• When k is 10
• When k is 100
• When k is 1000
• When k is 10000
• When k is 100000
• When k is 1000000

Data visualizations of the linear computation trajectory per iteration:

• When k is 10
• When k is 100
• When k is 1000
• When k is 10000
• When k is 100000
• When k is 1000000

Logs

(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 10


Start: 111097.0553196
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111097.3771835
Duration: 0.32186389999696985

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 100


Start: 111106.616347
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111107.1051824
Duration: 0.4888353999995161

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 1000


Start: 111174.1187486
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111174.482117
Duration: 0.3633684000087669

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 10000


Start: 111285.8601127
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111286.3067786
Duration: 0.44666590000269935

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 100000


Start: 111301.3935478
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111302.2409255
Duration: 0.8473776999890106

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler> python .\problems\problem_zero.py
When k = 1000000


Start: 111315.9840961
Heavy but optimised computation in progress...
Computation completed successfully. Writing benchmarks to file...
End: 111321.9413636
Duration: 5.957267499994487

All files saved successfully.
(.venv) PS C:\Users\HBRAI\Desktop\coding\project_euler>