Good introductory readings to topics related to prime numbers for non-mathematicians I'm a maths hobbyist who is fascinated by prime numbers. My quest to delve into the interesting parts of the topic is always hindered by my inability to understand the notation and concepts I encounter in published papers or any other mediums. I understand this problem is a result of my unfamiliarity with the areas involving the primes such as Number Theory, Sets, Logic and Reasoning, etc...
What are some graceful readings that one would recommend for me in order to get up to speed in said topics?
 A: The Art of Problem Solving (AOPS.org) has a website with abundant links to number theory resources,  (guides, exercises, reference recommendations) to begin studying number theory, at varying levels of difficulty - from very basic, to progressively more difficult. E.g., To start, see AOPS's Introductory Topics in Number Theory: 
"The following topics make a good introduction to number theory:"
Primes
    Sieve of Eratosthenes
    Prime factorization 
Composite numbers
Divisibility
    Divisors
        Common divisors
            Greatest common divisors 
        Counting divisors 
    Multiples
        Common multiples
            Least common multiples 
Division Theorem (the Division Algorithm)
Base numbers
Diophantine equations
    Simon's Favorite Factoring Trick 
Modular arithmetic
    Linear congruence 

(Links available for each topic above.)

Not necessarily directly related to number theory, but more foundational in nature to help prepare you for studying it, see the following
For some background reading in you listed topics: "sets, logic, and reasoning," and to better understand proof-reading/writing (with number theory as examples), you might want to start with 


*

*Velleman's How to Prove It: A Structured Approach.

*Mason, Burton, and Stacey's Thinking Mathematically
Together, they provide both structural knowledge (the "basics" of "building blocks"), and examples and exercises to get your "mind rolling" and to develop agility in your reasoning.
You can preview each of the books at the links above.
A: Since you are a hobbyist, I highly recommend "Number Theory Through Inquiry" by Marshall et al. It is a small book that presents elementary number theory in a way that you can explore the theorems yourself at your own pace. You don't need any math background to read and work through it. It may take a bit longer to go through than reading a normal textbook, but I find this style of learning way more enjoyable.
A: "A Friendly Introduction to Number Theory", by Joseph Silverman
"The Little Book of Big Prime Primes", by Paulo Ribenboim
"The New Book of Prime Number Records", by Paulo Ribenboim
"Prime Obsession", by John Derbyshire
"Prime Numbers", by David Wells
...to begin with.
