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# Question about number theory

Hi, now I am learning Number Theory and I have a few questions which I want to discuss. My first question is "Can anyone list topics which appear in Number Theory". I just know few topics like divisibility, some equations (Diophantine, Pell, ...), ... My second question is that in this link: http://www.artofproblemsolving.com/Forum/viewforum.php?f=131, I see the topic "Basic Algebraic Number Theory and applications" and "Analysis and Number Theory", are these also kinds of Number Theory or they are different? Thank you.

Note by Anh Huy Nguyen
4 years ago

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now quietly hear to me :::::::::::Number theory is a vast and fascinating field of mathematics, sometimes called (open curly double quote)higher arithmetic,(close curly double quote) consisting of the study of the properties of whole numbers. Primes and prime factorization are especially important in number theory, as are a number of functions such as the divisor function, Riemann zeta function, and totient function. Excellent introductions to number theory may be found in Ore and Beiler. The classic history on the subject (now slightly dated) is that of Dickson (2005abc). The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that (open curly double quote)it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics.(close curly double quote) Gauss, often known as the (open curly double quote)prince of mathematics,(close curly double quote) called mathematics the (open curly double quote)queen of the sciences(close curly double quote) and considered number theory the (open curly double quote)queen of mathematics(close curly double quote).......................... there are several topics:::::::::::::::::abstract algebra | additive number theory | algebraic number theory | analytic number theory | arithmetic | computational number theory | congruence | Diophantine equation | divisor function | elementary number theory | Gödel's first incompleteness theorem | Gödel's second incompleteness theorem | number theoretic function | Peano's axioms | prime counting function | prime factorization | prime number | quadratic reciprocity theorem | Riemann zeta function | totient function..................are some of them............1]Algebraic Integer 2] Dedekind Domain 3] Picard Group 4]Algebraic Number 5]Dedekind Ring 6] Pisot Constant 7]Algebraic Number Theory 8] Fractional Ideal 9]Pisot Number 10]Algebraics 11]Global Field 12] Pisot-Vijayaraghavan C... 13]Chebotarev Density The... 14] Kummer Extension Ramification Group 15] Class Field 16] Local Class Field Theory 17]Weyl Sum 18] Cyclotomic Field 19] Local Field 20] Decomposition Group 21]Number Field SignatureAlgebraic Integer 22] Gelfond-Schneider Theorem 23] Schanuel's Conjecture 24]Algebraic Number 25]Gelfond's Theorem 26] Shidlovskii Theorem 27]Algebraically Independent 28] Hermite-Lindemann Theorem 29] Six Exponentials Theorem 30]Constant Problem 31]Hermite's Theorem Thue Constant 32] E-Function 33] Lindemann-Weierstrass... 34] Thue-Morse Constant 35]Equality Testing 36] Liouville's Constant 37] Transcendental Number 38]Four Exponentials Conj... 39] Liouville Number 40] Uniformity Conjecture 40]Gelfond's Constant 42]Prime Algebraic Number 43]Zero Testing 44]Gelfond-Schneider Cons... 45]Radical Integer · 4 years ago

omg i laughed so hard at the (open curly double quote) · 4 years ago

Thank you so much, Sayan. But I'm still confused about the difference between "algebraic number theory", " analytic number theory", "additive number theory", " computational number theory". Can you explain more specific for me? Thank you. · 4 years ago