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# Conjectures 'R Us

## Contents

**Conjectures 'R Us** (called **CRUS** in short) was established in 2007 by Gary Barnes.

## Project definition

For every base (`b` ≤ 1030) for the forms `k`•`b`^{n}±1 there exists a unique value of `k` for each form that has been conjectured to be the lowest 'Sierpiński value' (+1 form) or 'Riesel value' (-1 form) that is composite for all values of `n` ≥ 1. `k`'s that have a trivial factor (one factor the same) for all `n`-values as well as `k`'s that make Generalized Fermat number's are not considered.

## Sub-project #1

Assist in proving the Liskovets-Gallot conjectures for the forms `k`•2^{n}±1 where `n` is always odd **and** where `n` is always even.

## Sub-project #2

Assist in proving the Sierpiński base 2 2nd conjecture for the form `k`•2^{n}+1.

The 1st conjectured `k` where all `n` are proven composite is `k`=78557 and is extensively tested by the PrimeGrid Prime Sierpiński Problem project.

The 2nd conjectured `k` where all `n` are proven composite is `k`=271129. The range of 78557 < `k` < 271129 has been extensively tested by the PrimeGrid Prime Sierpiński Problem and Extended Sierpiński Problem projects. All of these projects have omitted even `k`'s from testing. For the 1st conjecture there are no even `k`'s remaining. For the 2nd conjecture some even `k`'s remain. Therefore CRUS is testing even `k`'s for the Sierp base 2 2nd conjecture.

## Sub-project #3

Assist in proving the Riesel problem and prove the 2nd Riesel Problem for the form `k`•2^{n}-1.

The 1st conjectured `k` where all `n` are proven composite is `k`=509203 and is extensively tested by the PrimeGrid The Riesel Problem project.

The 2nd conjectured `k` where all `n` are proven composite is `k`=762701. The 1st conjecture project has omitted even k's from testing and some even k's remain. The 2nd conjecture has not previously been tested. Therefore CRUS Even Riesel is testing even `k`'s for the Riesel base 2 1st conjecture and the CRUS project is testing all `k`'s for the Riesel base 2 2nd conjecture.

## Goal

Prove the conjectures by finding at least one prime for all lower values of `k`. Many of the conjectures have already been proven but much more work is needed to prove additional bases. Proving them all is not possible but we aim to prove many of them.