Give reasons for all steps in a proof
... If integers m and n are both divisible by 3 then 1. n + m is also divisible by 3. 2. m•n is divisible by 9. 3. n2 + 3n is divisible by 9. Prop. 4: For any nonzero integer d, if integers m and n are both divisible by d, then m + n is also divisible by d. ...
... If integers m and n are both divisible by 3 then 1. n + m is also divisible by 3. 2. m•n is divisible by 9. 3. n2 + 3n is divisible by 9. Prop. 4: For any nonzero integer d, if integers m and n are both divisible by d, then m + n is also divisible by d. ...
Prime Factors - Skyline R2 School
... used. If you start with an odd number, you will not start with 2 ...
... used. If you start with an odd number, you will not start with 2 ...
PERFECT NUMBERS WITH IDENTICAL DIGITS Paul Pollack1
... 3. Proofs of Theorems 1 and 2 Proof of Theorem 1 Throughout this section we assume that g ≥ 2 is fixed. We begin by treating the case of even perfect numbers. Lemma 7. There are only finitely many repdigit numbers in base g which are even and perfect. In fact, all such numbers are strictly less than ...
... 3. Proofs of Theorems 1 and 2 Proof of Theorem 1 Throughout this section we assume that g ≥ 2 is fixed. We begin by treating the case of even perfect numbers. Lemma 7. There are only finitely many repdigit numbers in base g which are even and perfect. In fact, all such numbers are strictly less than ...
Prime Factorization
... A whole number that has exactly two unique factors, 1 and the number itself, is a prime number. A number greater than 1 with more than two factors is a composite number. Every composite number can be expressed as a product of prime numbers. This is called a prime factorization of the number. A ...
... A whole number that has exactly two unique factors, 1 and the number itself, is a prime number. A number greater than 1 with more than two factors is a composite number. Every composite number can be expressed as a product of prime numbers. This is called a prime factorization of the number. A ...
On the least common multiple of q
... An equivalent form of the prime number theorem states that log lcm(1, 2, . . . , n) ∼ n as n → ∞ (see, for example, [4]). Nair [7] gave a nice proof for the well-known estimate lcm{1, 2, . . . , n} ≥ 2n−1 , while Hanson [3] already obtained lcm{1, 2, . . . , n} ≤ 3n . Recently, Farhi [1] established ...
... An equivalent form of the prime number theorem states that log lcm(1, 2, . . . , n) ∼ n as n → ∞ (see, for example, [4]). Nair [7] gave a nice proof for the well-known estimate lcm{1, 2, . . . , n} ≥ 2n−1 , while Hanson [3] already obtained lcm{1, 2, . . . , n} ≤ 3n . Recently, Farhi [1] established ...
![arXiv:math/0412079v2 [math.NT] 2 Mar 2006](http://s1.studyres.com/store/data/013294887_1-d94fad656ee5fb5bde358f5c8c1d35cf-300x300.png)
Congruent Numbers and Heegner Points
... Both Monsky and Tian have proven their theorem based on the original method of Heegner. Heegner published his paper in 1952 as a 59 years old nonprofessional mathematician. In the same paper, Heegner solved Gauss’ class number one problem whose correctness was accepted by the math community only in ...
... Both Monsky and Tian have proven their theorem based on the original method of Heegner. Heegner published his paper in 1952 as a 59 years old nonprofessional mathematician. In the same paper, Heegner solved Gauss’ class number one problem whose correctness was accepted by the math community only in ...
Number Theory Questions
... or subtraction, for example 11 mod 2 = 1, which means “the remainder when we divide 11 by 2”. But usually the ‘mod’ comes at the end and in brackets. I use the equality symbol = from here on for convenience. This is one of the features of modulo arithmetic. For example, suppose we were to find the r ...
... or subtraction, for example 11 mod 2 = 1, which means “the remainder when we divide 11 by 2”. But usually the ‘mod’ comes at the end and in brackets. I use the equality symbol = from here on for convenience. This is one of the features of modulo arithmetic. For example, suppose we were to find the r ...