# Conjectures 'R Us

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

## Project definition

For every base (*b* ≤ 1030) for the forms *k*×*b*^{n}+1 and *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.

## Subproject #1

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

## Subproject #2

Assist in proving the Sierp 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 Seventeen or Bust 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 Prime Sierpinski Problem and Extended Sierpinski Problem projects but the projects have omitted even *k* 's from testing just like Riesel base 2 projects have. Therefore CRUS is testing even *k* 's for both the Riesel base 2 1st conjecture and Sierp 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.