Table Exponent Limits

For quite a while, the limits on each table had been set to 1024 bits. That is, each table's exponent limit had been set such that each full number, ignoring any factors (algebraic or otherwise), was less than 2¹⁰²⁴. The exception was numbers that have an Aurifeuillian factorization; in such cases it is the individual Aurifeuillian parts (the "L" and "M" numbers) which had been restricted to 1024 bits.

For reasons discussed in the final section of the page on Aurifeuillian factorization, this has led to problems in some tables, where the algebraic factors of the largest Aurifeuillian numbers were not reflected in the table. As shown there, the correct way to handle that was for the Aurifeuillian limit to be made smaller in those tables. Since that wasn't done initially, correcting the problem by lowering this limit would mean throwing out numbers which were already in the tables. Rather than doing this, we have instead elected to raise the exponent limit for non-Aurifeuillian numbers in those tables, to ensure that all algebraic factors are correctly reflected. In addition, if we raised this limit in one table, then for simplicity we also raised the limit in the corresponding table of opposite sign.

For example, we raised the non-Aurifeuillian limit in the 5ⁿ+3ⁿ table from 441 to 545, ensuring that all of the algebraic factors for Aurieuillean numbers up to 5¹⁶³⁵+3¹⁶³⁵ would be in the table. And so we also raised the limit in the 5ⁿ-3ⁿ table to have the two tables stay in sync.

This increase in limits affects the following tables:

With these changes, the exponent limits for each table are given below. Note that the limit shown in the column for "Aurifeuillian Extension" only affects one of the two tables in that row, whichever one has Aurifeuillian factors (see the page on Aurifeuillian factorization for details).