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SECTION 5-3
• Selected Topics from Number Theory
Slide 5-3-1
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SELECTED TOPICS FROM NUMBER THEORY
• Perfect Numbers• Deficient and Abundant Numbers• Amicable (Friendly) Numbers• Goldbach’s Conjecture• Twin Primes• Fermat’s Last Theorem
Slide 5-3-2
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PERFECT NUMBERS
Slide 5-3-3
A natural number is said to be perfect if it is equal to the sum of its proper divisors.
6 is perfect because 6 = 1 + 2 + 3.
8 is not because 8 1 + 2 + 4.
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DEFICIENT AND ABUNDANT NUMBERS
Slide 5-3-4
A natural number is deficient if it is greater than the sum of its proper divisors. It is abundant if it is less than the sum of its proper divisors.
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EXAMPLE: CLASSIFYING A NUMBER
Slide 5-3-5
Decide whether 12 is perfect, abundant, or deficient.
SolutionThe proper divisors of 12 are 1, 2, 3, 4, and 6. Their sum is 16. Because 16 > 12, the number 12 is abundant.
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AMICABLE (FRIENDLY) NUMBERS
Slide 5-3-6
The natural numbers a and b are amicable, or friendly, if the sum of the proper divisors of a is b, and the sum of the proper divisors of b is a.
The smallest pair of amicable numbers is 220 and 284.
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GOLDBACH’S CONJECTURE (NOT PROVED)
Slide 5-3-7
Every even number greater than 2 can be written as the sum of two prime numbers.
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EXAMPLE: EXPRESSING NUMBERS AS SUMS OF PRIMES
Slide 5-3-8
Write each even number as the sum of two primes.
a) 12 b) 40
Solution
a) 12 = 5 + 7
b) 40 = 17 + 23
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TWIN PRIMES
Slide 5-3-9
Twin primes are prime numbers that differ by 2.
Examples: 3 and 5, 11 and 13
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TWIN PRIMES CONJECTURE (NOT PROVED)
Slide 5-3-10
There are infinitely many pairs of twin primes.
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SEXY PRIME
• Sexy primes are prime numbers that differ from each other by six.
• For example, 5 and 11, 7 and 13, etc.• Find another sexy prime…• 11, 17• 17,23• 23, 29
Slide 5-3-11
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FERMAT’S LAST THEOREM
Slide 5-3-12
For any natural number n 3, there are no triples (a, b, c) that satisfy the equation
.n n na b c
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EXAMPLE: USING A THEOREM PROVED BY FERMAT
Slide 5-3-13
Every odd prime can be expressed as the difference of two squares in one and only one way.
Express 7 as the difference of two squares.
Solution2 27 16 9 4 3