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We can see 59% of the Moon’s surface from Earth. Also, number 59 corresponds to the last minute in an hour, and to the last second in a minute. Those are only some appearances of 59 in our daily basis.
A factor of a natural number is a positive divisor of the number. A proper factor of a natural number is a factor that is different from 1 and from the number itself. For example, $$95 = 5\times19 = 1\times95$$; thus, 1, 5, 19, and 95 are all factors of 95, but only 5 and 19 are proper factors of 95.
A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the three prime numbers below 59 are: 43, 47, and 53; and the three prime numbers above 59 are: 61, 67, and 71.
A composite number is a natural number that has proper factors. As we saw, 95 has two proper factors, thus it is a composite number.
Number 59 is prime because it doesn’t have proper factors. In other words, the only factors of 59 are 1 and itself. To be sure of it, we can use the following property.
If 𝒏 is a natural number, and neither of the prime numbers less than $$\sqrt n$$ divides 𝒏, then 𝒏 is a prime number.
Notice that 59<64, thus $$\sqrt{59}\;<\;\sqrt{64}\;=\;8$$. Therefore, the prime numbers less than $$\sqrt{59}$$ are 2, 3, 5 and 7. Moreover,
$$59 = (2\times29) + 1$$$$59 = (3\times19) + 2$$$$59 = (5\times11) + 4$$$$59 = (7\times8) + 3$$
Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 59. Then, by the property above, 59 is a prime number.
On the other hand, a prime number of objects can’t be arranged into a rectangular grid with more than one column and more than one row. This is another way of verifying that 59 is a prime number:
Fifty-nine is the 17th prime number, and 17 is also a prime number, thus 59 is a super-prime number: a prime that occupies a prime position in the list of all prime numbers.
Since 59 = 61 – 2, and 61 is also a prime number, then 59 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.
Fifty-nine can be classified into several classes of primes numbers. We will name three classes here, and then we will see to which of them 59 belongs.
$$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)+ 1 $$, or
$$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)- 1 $$ where
$$p_{1,\;\;}p_{2,\;\;}p_{3,\;…,}\;p_n\;$$ are the first n prime numbers.
$$p=2^n-1$$ where n is an integer.
Let’s find out why:
We invite you to read other articles on prime numbers, on our blog, to find out which other prime numbers belong to these classes.
Do you think 61 is a prime number?
Yes, because its only factors are 1 and itself.
No, because it doesn’t have proper factors.
No, because it is between (2 × 3 × 5) ± 1 and (2 × 3 × 5 × 7) ± 1.
No, because it is between $$2^5\;–\;1\;=\;31\; and\;2^6\;–\;1\;=\;63.$$
Yes, because 59=2(29)+1, where 29 is a prime number.
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