Diophantine Equations
... and Z = m2 + n2 . Clearly gcd(m, n) divides both X and Y , hence m and n are coprime; moreover, since X is odd, we have 1 ≡ X 2 = m2 − n2 mod 4, which implies that m is odd and n = 2k is even. Thus (Y /2)2 = mk with m and k coprime, hence m = a2 and k = b2 , giving X 2 = a4 − 4b4 . Now we repeat the ...
... and Z = m2 + n2 . Clearly gcd(m, n) divides both X and Y , hence m and n are coprime; moreover, since X is odd, we have 1 ≡ X 2 = m2 − n2 mod 4, which implies that m is odd and n = 2k is even. Thus (Y /2)2 = mk with m and k coprime, hence m = a2 and k = b2 , giving X 2 = a4 − 4b4 . Now we repeat the ...
solutions
... (1) Fermat repeatedly challenged English mathematicians by sending them problems he claimed to have solved and asking for proofs. Two of them were the following that he sent to Wallis: • Prove that the only solution of x2 + 2 = y 3 in positive integers is given by x = 5 and y = 3; • Prove that the o ...
... (1) Fermat repeatedly challenged English mathematicians by sending them problems he claimed to have solved and asking for proofs. Two of them were the following that he sent to Wallis: • Prove that the only solution of x2 + 2 = y 3 in positive integers is given by x = 5 and y = 3; • Prove that the o ...
Lecture Notes for MA 132 Foundations
... divisible by a prime. Any prime dividing n1 and n2 will also divide n0 . Once again, n0 both is and is not divisible by a prime. In either case, we derive an absurdity from the supposition that n0 is the least element of the set S. Since S ⊂ N, if not empty it must have a least element. Therefore S ...
... divisible by a prime. Any prime dividing n1 and n2 will also divide n0 . Once again, n0 both is and is not divisible by a prime. In either case, we derive an absurdity from the supposition that n0 is the least element of the set S. Since S ⊂ N, if not empty it must have a least element. Therefore S ...
Prime Numbers and the Convergents of a Continued Fraction
... of the data. There are many types of cryptosystems (see eg. [7], [8], [13]) in which their security relies on a computationally difficult mathematical problem. One such cryptosystem relies on a factoring problem to ensure security [7], and several factoring algorithms based on continued fractions ha ...
... of the data. There are many types of cryptosystems (see eg. [7], [8], [13]) in which their security relies on a computationally difficult mathematical problem. One such cryptosystem relies on a factoring problem to ensure security [7], and several factoring algorithms based on continued fractions ha ...
MATH 2400: PRACTICE PROBLEMS FOR EXAM 1 1) Find all real
... 2) a) Prove that 6 is an irrational number. You may use the fact that if an integer x2 is divisible by 6, then also x is divisible by 6. (For “extra credit”, prove the fact of the previous sentence using the uniqueness of prime factorizations.) Solution: let N be a positive integer. We claim that if ...
... 2) a) Prove that 6 is an irrational number. You may use the fact that if an integer x2 is divisible by 6, then also x is divisible by 6. (For “extra credit”, prove the fact of the previous sentence using the uniqueness of prime factorizations.) Solution: let N be a positive integer. We claim that if ...
Notes for 11th Jan (Wednesday)
... The previous proposition shows that indeed A has no largest number and B no smallest. Therefore, the rationals have gaps in between. The proposition also gives us an idea of how to correct them. Indeed, Definition : Let (S, ≤) be a totally ordered set (where a ≥ b means that b ≤ a). S is said to sa ...
... The previous proposition shows that indeed A has no largest number and B no smallest. Therefore, the rationals have gaps in between. The proposition also gives us an idea of how to correct them. Indeed, Definition : Let (S, ≤) be a totally ordered set (where a ≥ b means that b ≤ a). S is said to sa ...
LOWNESS NOTIONS, MEASURE AND DOMINATION
... / n∈ω UnA . This notion of randomness is often called Martin-Löf randomness (relative to A) or 1-randomness (relative to A). Definition 1.5 (Nies [14]). A ≤LR B if every B-random real is A-random. The idea of A ≤LR B is that A is no more useful than B in the sense that A does not “derandomize” any ...
... / n∈ω UnA . This notion of randomness is often called Martin-Löf randomness (relative to A) or 1-randomness (relative to A). Definition 1.5 (Nies [14]). A ≤LR B if every B-random real is A-random. The idea of A ≤LR B is that A is no more useful than B in the sense that A does not “derandomize” any ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.