Fermat near misses

The table below contains many examples of Fermat near misses, near-solutions of

xⁿ + yⁿ = zⁿ.

This extends the tables computed in 2000 by Noam Elkies (which cover z ≤ 2²³ ≈ 8.4·10⁶) up through z ≤ 10¹¹. For some background on how these were found, see this blog post or the original paper.

The rows are in fermat.csv. A much larger file, fermat-0.01.csv, has every result with |r| ≥ 0.01. Scatter plots of those points are interactive; a static copy is fermat-scatter.pdf.

Columns

• error: the relative error of the near-solution, (zⁿ − xⁿ − yⁿ) / zⁿ.
• r: a measure of the quality of a near-solution, r = n zⁿ⁻³ / (zⁿ − yⁿ − xⁿ). Results are included in the table if |r| ≥ 1.
• NDE: whether this near-solution is already reported in Noam D. Elkies' tables.
• Family: the near-solution belongs to one of the known families below.
• Notes: comments, for example when a near-solution is an integer multiple of another row.

Known families

• F4a: (192 t⁷)⁴ + (192 t⁸ − 24 t⁴ − 1)⁴ = (192 t⁸ + 24 t⁴ − 1)⁴ + 192 t⁴
• F4b: (6 t⁷)⁴ + (3 t⁸ − 6 t⁴ − 4)⁴ = (3 t⁸ + 6 t⁴ − 4)⁴ − 3072 t⁴
• F4c: (8192 t⁸ − 1)⁴ + (16384 t⁹ − 128 t⁵ − 3t)⁴ = (16384 t⁹ + 128 t⁵ − 3t)⁴ − (5120 t⁸ − 1)
• F4d: (64 t⁸ − 2)⁴ + (64 t⁹ − 8 t⁵ − 3t)⁴ = (64 t⁹ + 8 t⁵ − 3t)⁴ − 16(20 t⁸ − 1), for odd t
• F8a: (32 t⁸ − 7)⁸ + (32 t⁹ − 10 t)⁸ = (32 t⁹ − 6 t)⁸ + 21·2²⁸ t⁴⁰ + O(t³²)
• F8b: (32 t⁸ + 7)⁸ + (32 t⁹ + 6 t)⁸ = (32 t⁹ + 10 t)⁸ + 21·2²⁸ t⁴⁰ + O(t³²)
• F8c: (2⁵·7⁷ t⁸ − 1)⁸ + (2⁵·7⁸ t⁹ − 10 t)⁸ = (2⁵·7⁸ t⁹ − 6 t)⁸ + 2²⁸·3·7³³ t⁴⁰ + O(t³²)
• F8d: (2⁵·7⁷ t⁸ + 1)⁸ + (2⁵·7⁸ t⁹ + 6 t)⁸ = (2⁵·7⁸ t⁹ + 10 t)⁸ + 2²⁸·3·7³³ t⁴⁰ + O(t³²)

F4a and F8b are described in more detail in Elkies' notes. F8c and F8d arise from F8a and F8b when t is a multiple of 7: each term is then divisible by 7, and (x/7, y/7, z/7) is a new solution with the same relative error but a much higher r-score.

F4c and F4d come from (2t⁸ − 16)⁴ + (t⁹ − 2t⁵ − 12t)⁴ = (t⁹ + 2t⁵ − 12t)⁴ − 1024(5t⁸ − 64). Divide by 16 when t is a multiple of 4 to get F4c, and by 8 when t is twice an odd integer to get F4d. Either quotient has a much higher r-score. An even parameter of the F4d formula is twice an F4c solution, so those rows stay marked as multiples.

Table

Loading the table…