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Optimize nextprime by transitioning from 6n system to 30n system #26608

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References to other Issues or PRs

Brief description of what is fixed or changed

The nextprime function has been optimized by transitioning from the 6n system to the 30n system for prime candidate selection. This change reduces the number of unnecessary prime checks, improving the overall performance of the algorithm.

Other comments

In the previous implementation, potential primes were located at 6j + {1, 5}. By switching to the 30j + {1, 7, 11, 13, 17, 19, 23, 29} system, we eliminate two additional prime checks per cycle. ( For instance, 30n+5 is also 6n+5. Using this way we skip 30n+5 and 30n+25 ). This adjustment leverages a more efficient pattern for prime candidate selection, leading to faster computations.

Release Notes

  • ntheory
    • Optimized nextprime function by switching from 6n + {1,5} system to 30n + {1, 7, 11, 13, 17, 19, 23, 29} system, reducing unnecessary prime checks and improving performance.

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  • ntheory
    • Optimized nextprime function by switching from 6n + {1,5} system to 30n + {1, 7, 11, 13, 17, 19, 23, 29} system, reducing unnecessary prime checks and improving performance. (#26608 by @mthn1907)

This will be added to https://github.com/sympy/sympy/wiki/Release-Notes-for-1.13.

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#### References to other Issues or PRs
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#### Brief description of what is fixed or changed
The `nextprime` function has been optimized by transitioning from the 6n system to the 30n system for prime candidate selection. This change reduces the number of unnecessary prime checks, improving the overall performance of the algorithm.


#### Other comments
In the previous implementation, potential primes were located at 6j + {1, 5}. By switching to the 30j + {1, 7, 11, 13, 17, 19, 23, 29} system, we eliminate two additional prime checks per cycle. ( For instance, 30n+5 is also 6n+5. Using this way we skip 30n+5 and 30n+25 ). This adjustment leverages a more efficient pattern for prime candidate selection, leading to faster computations. 


#### Release Notes

<!-- Write the release notes for this release below between the BEGIN and END
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and the release note for this PR. For example:

* solvers
  * Added a new solver for logarithmic equations.

* functions
  * Fixed a bug with log of integers. Formerly, `log(-x)` incorrectly gave `-log(x)`.

* physics.units
  * Corrected a semantical error in the conversion between volt and statvolt which
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<!-- BEGIN RELEASE NOTES -->
* ntheory
  * Optimized `nextprime` function by switching from 6n + {1,5} system to 30n + {1, 7, 11, 13, 17, 19, 23, 29} system, reducing unnecessary prime checks and improving performance. 
<!-- END RELEASE NOTES -->

@mthn1907 mthn1907 closed this May 18, 2024
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