Manuscript Oai119-A@1 · Oai119-A.tex · lemma · fdff7292

Lemma: Root enclosures

Stages 3 and 6: the complete finite certificate › Exact integer-square certificates · lines 868–897 · proof included or linked

Oai119-A@1#lem:roots

Let $Q = 100000$ . The following strict enclosures hold:

\frac{173205}{Q} < 3 < \frac{173206}{Q}, \frac{264575}{Q} < 7 < \frac{264576}{Q},

\frac{387298}{Q} < 15 < \frac{387299}{Q}, \frac{624499}{Q} < 39 < \frac{624500}{Q} .

Proof. For each row of the following table, both displayed residuals are positive integers:

k 371539 a 173205264575387298624499 b 173206264576387299624500 k Q^{2} - a^{2} 2797569375259196998999 b^{2} - k Q^{2} 318436459776515401250000 .

Hence $a^{2} < k Q^{2} < b^{2}$ . Taking nonnegative square roots and dividing by $Q$ proves the result.

Displayed formulas in this passage

unlabeled · line 871
$\frac{173205}{Q} < 3 < \frac{173206}{Q}, \frac{264575}{Q} < 7 < \frac{264576}{Q},$
unlabeled · line 876
$\frac{387298}{Q} < 15 < \frac{387299}{Q}, \frac{624499}{Q} < 39 < \frac{624500}{Q} .$
unlabeled · line 886
$k 371539 a 173205264575387298624499 b 173206264576387299624500 k Q^{2} - a^{2} 2797569375259196998999 b^{2} - k Q^{2} 318436459776515401250000 .$