Euc20205/Sub-Problem 1: Difference between revisions
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==Problem== | |||
(a) For each positive real number x, define <math>f(x)</math> to be the number of prime numbers | (a) For each positive real number x, define <math>f(x)</math> to be the number of prime numbers | ||
<math>p</math> that satisfy <math>x \le p \le x + 10</math>. What is the value of <math>f(f(20))</math>? | <math>p</math> that satisfy <math>x \le p \le x + 10</math>. What is the value of <math>f(f(20))</math>? | ||
==Solution== | |||
Because <math>20</math> is a relatively small number, we can just bash this out. We first need to calculate <math>f(20)</math>, so the numbers in this interval are <math>20</math> to <math>30</math>. The only prime numbers in this range are <math>23</math> and <math>29</math>, so <math>f(20)=2</math>. Then, we need to find <math>f(2)</math>. The range we have is <math>2</math> to <math>12</math>, and the prime numbers that are in this interval are <math>2</math>, <math>3</math>, <math>5</math>, <math>7</math>, and <math>11</math>, so <math>f(2)=\boxed{5}</math>. | |||
~Baihly2024 | |||
Latest revision as of 17:05, 12 October 2025
Problem
(a) For each positive real number x, define 
to be the number of prime numbers
that satisfy 
. What is the value of 
?
Solution
Because 
is a relatively small number, we can just bash this out. We first need to calculate 
, so the numbers in this interval are to 
. The only prime numbers in this range are 
and 
, so 
. Then, we need to find 
. The range we have is 
to 
, and the prime numbers that are in this interval are , 
, 
, 
, and 
, so 
.
~Baihly2024
