Determine whether m_13 = 2^13 - 1 = 8191 and m_23 = 2^23 -1 = 8388607 are prime.© BrainMass Inc. brainmass.com October 25, 2018, 12:40 am ad1c9bdddf
You can use Fermat's little theorem, which states that:
y^(q - 1) Mod q = 1
where q is a prime and y has no divisor in common with q (we can also say that y is not equal to zero modulo q) . Let me know if you are not familiar with modular arithmetic and don't know what "mod" means.
Suppose that we have some Mersenne number
M_p = 2^p - 1
with p an odd prime. Also, we assume that M_p is not prime. Then we use Fermat's little theorem to find a constraint that any prime divisors of M_p must satisfy. Before we derive this, let's first explain how we are going to use that constraint. We compute M_p Modulo the primes satisfying the constraint to see if one of them divides M_p. We know that:
M_p is not prime -----> There is a prime satisfying the constraint that divides M_p
This is equivalent to:
NOT[There is a prime satisfying the constraint that divides M_p] ----------> NOT[M_p is not prime]
which we can rewrite as:
No primes satisfying the constraint divide M_p ...
A detailed solution explaining Euler's method of primality testing for Mersenne and Fermat numbers is given.
Perfect and amicable numbers
1) See attached
2) Show that 28 is a perfect number.
3) What is the amicable number to 17296?
4) Express all 5/11 as the sum of distinct unit fractions.
5) Using modular arithmetic, derive a formula to find all numbers that meet the following conditions:
- Divide by 3 the remainder is 1
- Divide by 5 the remainder is 0
6) What is the area of a triangle with lengths of sides 28, 30 and 32?
7) What is the probability of picking 6 numbers out of 53 numbers and getting
4 out of 6 numbers correct?
8) Lagrange contributed a great deal to the metric system. How would you show your students how to use the metric highway below?
9) See attached
10) See attached
11) Chevalier de Mere's Problem-
How many times must you throw two die in order to have half a chance of getting double sixes? (The original problem used the word dice but die is singular and dice is plural i.e., a pair of dice is two die.)
We know that (see attached). If you roll dice you have 36 outcomes. Also the probability of getting at least one double 6 on n rolls of the dice is the same as getting no double sixes on n rolls of a dice.
The probability of double sixes is 1/36. The probability of any combination other than double sixes is 35/36.
Thus, Pascal and Fermat concluded that solving for n in the formula below will solve the problem.
What is the minimum number of rolls of the dice to have at least 1/2 chance of getting double sixes?
12) Explain how to multiply 88 times 112 using Egyptian multiplication.
13) What 3 expressions can generate all Pythagorean triples.
14) Find an arithmetical progression with 5 terms, sum 11, and common difference 1/2.View Full Posting Details