15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for n = 434 and 8575.
15/If n and n +2 are a pair of twin primes, establish that o (n +2) = 0 (n) +2; this also holds for n = 434 and 8575.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
15
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NUMBER-THEORETIC FUNCTIONS
111
ton a positive integer k > I, show that there are infinitely many integers n for which
ln) = k, but at most finitely many n with o (n) = k.
%3D
Find the form of all positive integers n satisfying t(n) = 10. What is the smallest
) Show that there are no positive integers n satisfying o (n) = 10.
[Hint: Note that for n > 1, o (n) > n.]
1a Prove that there are infinitely many pairs of integers m and n with o (m2) = o (n²).
THint: Choose k such that gcd(k, 10) = 1 and consider the integers m = 5k, n = 4k.]
ST
%3D
(a) n = 2k-1 satisfies the equation o (n) = 2n – 1.
(6) If 2* – 1 is prime, then n = 2k-'(2k - 1) satisfies the equation o (n) = 2n.
(c) If 2k – 3 is prime, then n = 2k-1(2k – 3) satisfies o (n) = 2n + 2.
It is not known if there are any positive integers n for which o (n) = 2n + 1.
15/If n and n + 2 are a pair of twin primes, establish that o (n +2) = o (n) + 2; this also
%3D
%3D
%3D
holds for n = 434 and 8575.
16. (a) For any integer n > 1, prove that there exist integers n1 and n2 for which
T(n1) + t(n2) = n.
(b) Prove that the Goldbach conjecture implies that for each even integer 2n there exist
integers n1 and n2 with o (n1) +o (n2) = 2n.
17. For a fixed integer k, show that the function f defined by f(n) = nk is multiplicative.
18. Let f and g be multiplicative functions that are not identically zero and have the property
that f(pk) = g(pk) for each prime p and k > 1. Prove that f = g.
19. Prove that if f and g are multiplicative functions, then so is their product fg and quotient
f/g (whenever the latter function is defined).
20. Let w(n) denote the number of distinct prime divisors of n > 1, with w(1) = 0. For
instance, w(360) = @(2³ . 32 . 5) = 3.
(a) Show that 2(n) is a multiplicative function.
(b) For a positive integer n, establish the formula
%3D
T(n²) = 20d)
LEGO
u|p
21. For any positive integer n, prove that Ean T(d)³ = (Ea\n T(d))*.
[Hint: Both sides of the equation in question are multiplicative functions of n, so that it
suffices to consider the case n =
pk, where p is a prime.]
22. Given n > 1, let o, (n) denote the sum of the sth powers of the positive divisors of n3;
that is,
%D
ulp
Verify the fowing:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2Fc8e6d305-e9ee-4b4e-9943-98743c1fa2ee%2F4vx97ho.jpeg&w=3840&q=75)
Transcribed Image Text:hat
NUMBER-THEORETIC FUNCTIONS
111
ton a positive integer k > I, show that there are infinitely many integers n for which
ln) = k, but at most finitely many n with o (n) = k.
%3D
Find the form of all positive integers n satisfying t(n) = 10. What is the smallest
) Show that there are no positive integers n satisfying o (n) = 10.
[Hint: Note that for n > 1, o (n) > n.]
1a Prove that there are infinitely many pairs of integers m and n with o (m2) = o (n²).
THint: Choose k such that gcd(k, 10) = 1 and consider the integers m = 5k, n = 4k.]
ST
%3D
(a) n = 2k-1 satisfies the equation o (n) = 2n – 1.
(6) If 2* – 1 is prime, then n = 2k-'(2k - 1) satisfies the equation o (n) = 2n.
(c) If 2k – 3 is prime, then n = 2k-1(2k – 3) satisfies o (n) = 2n + 2.
It is not known if there are any positive integers n for which o (n) = 2n + 1.
15/If n and n + 2 are a pair of twin primes, establish that o (n +2) = o (n) + 2; this also
%3D
%3D
%3D
holds for n = 434 and 8575.
16. (a) For any integer n > 1, prove that there exist integers n1 and n2 for which
T(n1) + t(n2) = n.
(b) Prove that the Goldbach conjecture implies that for each even integer 2n there exist
integers n1 and n2 with o (n1) +o (n2) = 2n.
17. For a fixed integer k, show that the function f defined by f(n) = nk is multiplicative.
18. Let f and g be multiplicative functions that are not identically zero and have the property
that f(pk) = g(pk) for each prime p and k > 1. Prove that f = g.
19. Prove that if f and g are multiplicative functions, then so is their product fg and quotient
f/g (whenever the latter function is defined).
20. Let w(n) denote the number of distinct prime divisors of n > 1, with w(1) = 0. For
instance, w(360) = @(2³ . 32 . 5) = 3.
(a) Show that 2(n) is a multiplicative function.
(b) For a positive integer n, establish the formula
%3D
T(n²) = 20d)
LEGO
u|p
21. For any positive integer n, prove that Ean T(d)³ = (Ea\n T(d))*.
[Hint: Both sides of the equation in question are multiplicative functions of n, so that it
suffices to consider the case n =
pk, where p is a prime.]
22. Given n > 1, let o, (n) denote the sum of the sth powers of the positive divisors of n3;
that is,
%D
ulp
Verify the fowing:
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