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if n is a natural number such that n=p1 power a1 *p2 power a2*p3 power a3..............*pk power ak and pk are distinct primes , then show that log n>=k log2

if n is a natural number such that n=p1 power a1 *p2 power a2*p3 power a3..............*pk power ak and pk are distinct primes , then show that log n>=k log2


Grade:11

1 Answers

Suyash Mishra
15 Points
2 years ago
n= p(1)^a(1) + p(2)^a(2) + p(3)^a(3) + .........+ p(k)^a(k)
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n is a natural number,,,this leads to n=1 or larger integer. ,,,, AND it limits each term of the sequence to a positive number. That is, the p"s cannot be + or -, as this would restrict the terms to an even number, to make n positive, which was not stated.
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Likewise, the "a" values cannot be negative, leading to a fraction, or a fraction, which would lead to not whole numbers.
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The given that p(k) is a distinct prime, leads to the min value of p(k) being the min prime, or 2 .,,,or larger,,, (1 is a special number, not prime)
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n = something positive * a number 2 or larger raised to a positive number( making it >2)
.
n>2^ k
.
and taking log of both sides
.
log n > log 2^k or log n > k log 2
 
 Answer By Suyash Mishra Kanpur

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