Modern Algebra

1

We first tabulate (n2 - 1 ) for n 2 to 15 and try to look for some patterns

n (n2 - 1 ) Prime

2 3 YES

3 8 no

4 15 no

5 24 no

6 35 no

7 48 no

8 63 no

9 80 no

10 99 no

11 120 no

12 143 no

13 168 no

14 195 no

15 224 We see that (n2-1 ) is a prime only when n 2 . We then make the following conjecture (n2-1

) is a prime number if and only if n 2

We attempt to prove this conjecture below . We begin by factoring (n2 -1 as the difference of two squares Therefore , any (n2 - 1 ) has two integer factors (n 1 ) and (n-1 Clearly (n 1 ) is greater than (n - 1 . Additionally , any prime number only has two possible factors , 1 and itself . For (n2 - 1 ) to be a prime its two factors (n 1 ) and (n - 1 ) will have to equal (n2 - 1 ) and 1 respectively . This is only possible when n 2 . Thus we prove our conjecture

2

For an E-number to be an E-prime , if we factor the E-number using only elements of the E-number system , the only factors are 1 and the E-number itself

Similarly , for an E-number to be an E-composite , it is possible to create a factorization of the number aside from the E-number and 1 , that contains only numbers which are present in the E-number system . We should also note that except for the special number 1 , the condition of E-prime and E-composite are distinct . An E-number can either be a E-prime or E-composite but not both . Additionally , all E-numbers except 1 belong to either set , there is no E-number that does not belong in the E-prime and E-composite sets . With this definition , the first 10 E-numbers are the following (if we do not consider 1

2 , 6 , 10 , 12 , 14 , 18 , 22 , 26 , 30 , and 34

An E-composite number whose E-prime factorization is not unique is 72 The possible E-prime factorizations of 72 are the following

72 2 x 2 x 18 and 72 12 x 6

Let us assume that E-composites exist which cannot be written as the product of E-primes . Then there is a number n greater than 1 which is the smallest E-composite E-number that cannot be written as the product of two E-primes . Since n is an E-composite , we can write n as the following

n ab

where a and b are E-numbers too . Also note that both a and b are smaller than n . Since n cannot be expressed as the product of E-primes both a and b cannot have E-prime factors . However , n is the smallest possible E-number which does not have E-prime factors . This presents a contradiction thus our assumption , that E-composites which cannot be written as the product of E-primes , is false

3

We perform the Euclidean algorithm to get the Greatest Common Divisor (GCD ) of 231 and 168 using the table method

A Q...