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... will be used for the least common multiple of ai, a ..., a t . As will be seen, the ^-function is useful for generating sequences of rational numbers which are used to construct generalized Kummer congruences. This paper is concerned with sequences {UJ}-Q of rational numbers. It will be supposed ...
... will be used for the least common multiple of ai, a ..., a t . As will be seen, the ^-function is useful for generating sequences of rational numbers which are used to construct generalized Kummer congruences. This paper is concerned with sequences {UJ}-Q of rational numbers. It will be supposed ...
MIXED SUMS OF SQUARES AND TRIANGULAR NUMBERS (III)
... 4 if and only if p2 = x2 + 8(y 2 + z 2 ) for no odd integers x, y, z. (ii) Let n > 1 be an odd integer. Then all prime divisors of n are congruent to 1 mod 4, if and only if n2 = x2 + 4(y 2 + z 2 ) for no odd integers x, y, z. Remark 1.2. In number theory there are very few simple characterizations ...
... 4 if and only if p2 = x2 + 8(y 2 + z 2 ) for no odd integers x, y, z. (ii) Let n > 1 be an odd integer. Then all prime divisors of n are congruent to 1 mod 4, if and only if n2 = x2 + 4(y 2 + z 2 ) for no odd integers x, y, z. Remark 1.2. In number theory there are very few simple characterizations ...
Transcendental nature of special values of L-functions
... Weak Schanuel Conjecture Let α1 , . . . , αn be non-zero algebraic numbers such that log α1 , . . . , log αn are linearly independent over Q . Then these numbers are algebraically independent. We shall need the following important consequence of the Weak Schanuel Conjecture. Proposition 2.3 Assume t ...
... Weak Schanuel Conjecture Let α1 , . . . , αn be non-zero algebraic numbers such that log α1 , . . . , log αn are linearly independent over Q . Then these numbers are algebraically independent. We shall need the following important consequence of the Weak Schanuel Conjecture. Proposition 2.3 Assume t ...
... c) A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water. At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out? d) An observer is tracking a plane flying at an altitude of 5000 ft. The plane flies directly ov ...
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... N (d) = {n : Fn is elliptic Korselt for Q( −d)}. 2 , it follows that if r ≥ 5 It is easy to prove that N (1) = ∅. Namely, since F2n+1 = Fn2 + Fn+1 is an odd prime, then all prime factors of Fr are congruent to 1 modulo 4. In particular, (−1|p) = 1 for all prime factors p of Fr . Since Fr | Fn for al ...
... N (d) = {n : Fn is elliptic Korselt for Q( −d)}. 2 , it follows that if r ≥ 5 It is easy to prove that N (1) = ∅. Namely, since F2n+1 = Fn2 + Fn+1 is an odd prime, then all prime factors of Fr are congruent to 1 modulo 4. In particular, (−1|p) = 1 for all prime factors p of Fr . Since Fr | Fn for al ...
Chapter 2 Lesson 1: Rational Numbers
... cream. A dish with two scoops can have any two flavors, including the same flavor twice. How many different double-scoop combinations are possible? ...
... cream. A dish with two scoops can have any two flavors, including the same flavor twice. How many different double-scoop combinations are possible? ...
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... To complete the proof, we will show all the roots are distinct except when n = 0 (mod 3). In that case, x = - 1 will be a double root. To verify this, first observe that by (2) and (4) x - -1 is a root of Pn(x) if and only if n = 0 (mod 3). Now the derivative of P„(x) is P;(x) = ...
... To complete the proof, we will show all the roots are distinct except when n = 0 (mod 3). In that case, x = - 1 will be a double root. To verify this, first observe that by (2) and (4) x - -1 is a root of Pn(x) if and only if n = 0 (mod 3). Now the derivative of P„(x) is P;(x) = ...
9.6 Mathematical Induction
... The words induction and deduction are usually used to contrast two patterns of logical thought. We reason by induction when we use evidence derived from particular examples to draw conclusions about general principles. We reason by deduction when we reason from general principles to draw conclusions ...
... The words induction and deduction are usually used to contrast two patterns of logical thought. We reason by induction when we use evidence derived from particular examples to draw conclusions about general principles. We reason by deduction when we reason from general principles to draw conclusions ...
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... So if each factor in the standard form of a number is representable, then the number itself is representable. Once again, let us examine the standard form of a number N ; here, however, N is completely arbitrary. Trivially, 2 = 12 + 12 + 02 + 02 , so any power of 2 is representable. It immediately f ...
... So if each factor in the standard form of a number is representable, then the number itself is representable. Once again, let us examine the standard form of a number N ; here, however, N is completely arbitrary. Trivially, 2 = 12 + 12 + 02 + 02 , so any power of 2 is representable. It immediately f ...
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... "Prime Factorization" is finding which prime numbers multiply together to make the original number. Here are two examples: 1) What is the prime factorization of 147 ? Can we divide 147 evenly by 2? ...
... "Prime Factorization" is finding which prime numbers multiply together to make the original number. Here are two examples: 1) What is the prime factorization of 147 ? Can we divide 147 evenly by 2? ...
