Can a Natural Number Satisfy n ≡ 1 (mod p) for All Primes in a Large Set?

In summary, "Numbers in the same class" refer to a group of numbers that share similar characteristics or properties. The criteria for determining which numbers belong in the same class can vary, and studying numbers in the same class can help us identify patterns and relationships between numbers. Numbers can belong to multiple classes at the same time, and numbers in the same class are related because they share common characteristics and can be compared to each other.
  • #1
Bibubo
14
0
Let $\left\{ p_{1},p_{2},\dots,p_{h}\right\}$ a set of consecutive prime numbers. I want to show that, if $h$ is large enough, then doesn't exists a natural number $n$ such that $$n\equiv1\textrm{ mod }p_{i},\,\forall i=1,\dots,h.$$
I think is true but I have no idea how to prove it. Am I wrong?
 
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  • #2
Your claim is unfortunately false. Try $n = p_1 p_2 \cdots p_h + 1$ (and the primes don't need to be consecutive either).
 

FAQ: Can a Natural Number Satisfy n ≡ 1 (mod p) for All Primes in a Large Set?

What are "numbers in the same class"?

"Numbers in the same class" refer to a group of numbers that share similar characteristics or properties, such as being even or odd, prime or composite, or divisible by a common number.

How do you determine which numbers belong in the same class?

The criteria for determining which numbers belong in the same class can vary depending on the specific characteristics or properties being considered. For example, to determine if numbers are even or odd, we look at whether they are divisible by 2. To determine if numbers are prime or composite, we check if they have any factors other than 1 and themselves.

What is the significance of studying numbers in the same class?

Studying numbers in the same class can help us identify patterns and relationships between numbers, as well as make predictions and solve problems more efficiently. It can also provide a deeper understanding of the fundamental properties and behaviors of numbers.

Can numbers belong to more than one class?

Yes, numbers can belong to multiple classes at the same time. For example, the number 6 belongs to the class of even numbers, but it also belongs to the class of numbers divisible by 3. This is because 6 is both even and divisible by 3.

How are numbers in the same class related to each other?

Numbers in the same class share common characteristics or properties, which means they can be grouped together and compared to one another. This can help us understand how different types of numbers behave and interact with each other.

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