Math Is Fun Forum

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There are several other instances where I get logged out as well.

I think you may be underestimating yourself a little. The only preparation you need for Spivak/Apostol is Lang's "Basic Mathematics" along with some precalculus.

If you are looking for an algebra book, Gelfand and Sullivan are two good options. You would be hard pressed to find an algebra book that is in the form Definition-Theorem-Proof (I don't see why you would want to either) but those two books proof enough theorems.

Going through a whole book on proofs might be a little too much. Just search some basic notes on it online. However, if you do need a book somewhere down the line, a great one is "How to Prove It: A Structured Approach" by Daniel Velleman.

By the way, you might also want to check out Apostol's calculus book. In some aspects, it is a little better than Spivak. For example, Apostol covers Linear Algebra, puts an emphasis on history and covers integral calculus before differential calculus.

Assume that the Goldbach conjecture is true (to the limit in the problem).

Now, break it into cases:

N < 2K: The answer is “No”, because the sum of K primes is at least 2K.
N ≥ 2K and K = 1: The answer is “Yes” if N is prime, and “No” otherwise.
N ≥ 2K, K = 2 and N is even: The answer is “Yes” (by Goldbach’s conjecture).
N ≥ 2K, K = 2 and N is odd: The answer is “Yes” if N − 2 is prime, and “No” otherwise. This is because the sum of two odd primes is even, and the only even prime number is 2, so one of the prime numbers in the sum must be 2.
N ≥ 2K and K ≥ 3: The answer is “Yes”. This is because if N is even, then N − 2(K − 2) is also even, so it is the sum of two primes, say p and q (by Goldbach’s conjecture). Thus, N is the sum of the K primes 2, 2, 2, ..., p, q. And if N is odd, then N − 3 − 2(K − 3) is even, so it is the sum of two primes, say p and q. Thus, N is the sum of the K primes 3, 2, 2, ..., p, q.

This code uses the Rabin-Miller algorithm and Goldbach conjecture

I believe that this solution works (it is in Python 3)

import math, random, sys


def is_probable_prime(n, k = 7):
    
    if n < 6:  
        return [False, False, True, True, False, True][n]
    
    elif n & 1 == 0:
        return False
    
    else:
        s, d = 0, n - 1

        while d & 1 == 0:
            s, d = s + 1, d >> 1

    for a in random.sample(range(2, n - 2), min(n - 4, k)):
        x = pow(a, d, n)

        if x != 1 and x + 1 != n:
            for r in range(1, s):
                x = pow(x, 2, n)

                if x == 1:
                    return False  
                elif x == n - 1:
                    a = 0  
                    break  
        
            if a:
                return False  

    return True


if __name__ == '__main__':
    T = int(sys.stdin.readline())

    
    for _ in range(T):
        N, K = list(map(int, sys.stdin.readline().split()))

        
        if N < 2 * K:
            print('No')
        
        elif K == 1:
            print('Yes' if is_probable_prime(N) else 'No')
        
        elif K == 2:
            if N % 2 == 0:
                print('Yes')
            else:
                print('Yes' if is_probable_prime(N - 2) else 'No')

        else:
            print('Yes')

What is the question?

While it is not true that you should be in academia to publish papers, there are some drawbacks. Authors writing by themselves are welcome, but they somehow are not very common because they lack the ability of writing an academic paper: they do not lack the message or the content of a paper, but just the structure, and that's something that you learn in an academic environment.  One famous example is Paul Erdos ( http://en.wikipedia.org/wiki/Paul_Erd%C5%91s#Career ), who I believe had no affiliation (at least in practice) for much of his life. However, most cultures have certain standard ways of doing things. Academia, and in particular the part of it that you want publish in, is no exception. One thing you can do to help get your ideas accepted is to learn to write and talk in the language common to your research area. Specifically, find some papers in your area that you really like (even better if they are widely cited) and study how they are written. When you write your own papers, make a conscious effort to copy the writing style of the papers you like. One key part of this is to thoroughly know the relevant literature (previous work on the problem) and to mention it in your introduction and explain how your work relates to it. For each research area, there are numerous other hurdles you should jump. For example, if you're writing a math paper, do it in LaTeX. If you don't know LaTeX, learn it (ask Google, if you need help), since using it will make your paper more readily accepted. For a list of other criteria often used by mathematicians to quickly judge a paper, read http://www.scottaaronson.com/blog/?p=304 .

If you are clueless in knowing where to publish your work, then I am guessing that you are also not familiar with all the relevant mathematics journals out there. This then leads me to conclude that you haven't been doing any sort of an extensive literature search on the state of knowledge of whatever area it is that you did your "independent research" on.

So my questions to you are:

1. How would you know your work is "valid" and free of errors and misunderstanding? The validity would come from self-consistency with the existing theories, and from experimentally verified observations.

2. How would you know that what you have done has not been published already, or has already been disproven?

Please note that the probability of someone without a formal education in mathematics, coming up with something "new" on any topic in mathematics, is practically zero. Don't believe me, just try finding the last time this has happened. Unless you believe that you have an unusually exceptional quality, insight, and abilities, I strongly suggest you do not fall into such illusion and make sure first that what you have is halfway decent. Find someone who has a mathematics qualification and ask him/her to review what you have done. Even Nobel Prize laureates give their manuscripts to someone else for a 2nd opinion, so why not you?

Although practically all parts help in understanding an important moral value.

Yeah I think he said that because he isn't allowed to be rude.

Welcome to the forum!

That is the right book (by the way, it is available online and legally at https://archive.org/details/Calculus_643). Although, I'm pretty surprised you're covered precalculus/algebra already. However, even if you have a few gaps, you can fill them up later.

As for the problems, they are basically a must if you want to gain a good level of understanding. They're half the reason why Spivak is such a good book. Also, even if you put in about 8 hours a day, Spivak in 2 weeks is pretty ambitious.