1.4 KiB
| id | title | challengeType | forumTopicId | dashedName |
|---|---|---|---|---|
| 5900f4ab1000cf542c50ffbd | Problem 318: 2011 nines | 5 | 301974 | problem-318-2011-nines |
--description--
Consider the real number √2+√3.
When we calculate the even powers of √2+√3
we get:
(√2+√3)2 = 9.898979485566356...
(√2+√3)4 = 97.98979485566356...
(√2+√3)6 = 969.998969071069263...
(√2+√3)8 = 9601.99989585502907...
(√2+√3)10 = 95049.999989479221...
(√2+√3)12 = 940897.9999989371855...
(√2+√3)14 = 9313929.99999989263...
(√2+√3)16 = 92198401.99999998915...
It looks like that the number of consecutive nines at the beginning of the fractional part of these powers is non-decreasing. In fact it can be proven that the fractional part of (√2+√3)2n approaches 1 for large n.
Consider all real numbers of the form √p+√q with p and q positive integers and p<q, such that the fractional part of (√p+√q)2n approaches 1 for large n.
Let C(p,q,n) be the number of consecutive nines at the beginning of the fractional part of (√p+√q)2n.
Let N(p,q) be the minimal value of n such that C(p,q,n) ≥ 2011.
Find ∑N(p,q) for p+q ≤ 2011.
--hints--
euler318() should return 709313889.
assert.strictEqual(euler318(), 709313889);
--seed--
--seed-contents--
function euler318() {
return true;
}
euler318();
--solutions--
// solution required