Circulant-determinant Orders 3 and 4

Enter a, b, c to compute the circulant determinant and find every elegant identity for it.

For a 4×4 circulant matrix with first row (a,b,c,d), the rows are [a,b,c,d], [d,a,b,c], [c,d,a,b], [b,c,d,a].
The determinant always factors as (a+b+c+d)·(a−b+c−d)·[(a−c)²+(b−d)²]. This tool also checks for elegant coincidental matches — square+sum, negated+product, squared signed-sum, and scaling identities — just like the order-3 tab.

Same search as tab ⑤, but collapses every (a,b,c,d) that's just a row-rotation or an axis-reflection of another entry — same circulant matrix, same determinant, different starting point — into a single canonical result. Nothing in tabs ①–⑤ is affected by this.

Same search as tab ①, but collapses every permutation of (a,b,c) into a single canonical result. Unlike order 4, a³+b³+c³-3abc is fully symmetric — all 6 orderings of a triple give the identical determinant, not just the 3 cyclic rotations — so this tab dedupes more aggressively than tab ⑥ does. Nothing in tabs ①–⑥ is affected by this.

Instead of searching triples and hoping some land on an AP, this constructs them directly: pick a middle term b and a common difference e, and the triple (b−e, b, b+e) is guaranteed arithmetic. Every result here is exact — no coincidence, no search — because the identity a³+b³+c³-3abc = 9e²b always holds for an AP.

For order 4, choose a first term b and a common difference e. The first row (b, b+e, b+2e, b+3e) is then an ordinary arithmetic progression. Its circulant determinant always satisfies det = −16e³(a+b+c+d). Every result below is exact.

Order 3, self-contained: search a range, verify a single triple, and download results — all in one tab, the same layout as tab ⑤. Tabs ①–④ are untouched; this is an additional, uniform way to reach the same order-3 logic.

Run a search in tab ① first. The TXT file follows the selected display mode:
Grouped by type creates separate sections for every identity mode; No — list all creates one complete sequential list.

Run a search in tab ① first. The Excel workbook follows the selected display mode:
Grouped by type creates a summary plus a separate worksheet for every identity mode; No — list all creates one complete results worksheet.