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**1 - 8**of**8**### Some Information-Theoretic Computations . . .

- SUBMITTED TO JORMA RISSANEN’S FESTSCHRIFT VOLUME
, 2008

"... We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all prim ..."

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We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finite-n bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.

### LE MATEMATICHEVol. LVII (2002) Fasc. I, pp. 111117 A NOTE ON TWIN PRACTICAL NUMBERS

"... A positive integer m is a practical number if every positive integern < m is a sum of distinct divisors of m. Let P2(x) be the counting functionof the pairs (m,m + 2) of twin practical numbers. Margenstern conjecturedthat P2(x) ∼ λ2x(log x)−2. We prove that, for suf�ciently large x and for asuit ..."

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A positive integer m is a practical number if every positive integern < m is a sum of distinct divisors of m. Let P2(x) be the counting functionof the pairs (m,m + 2) of twin practical numbers. Margenstern conjecturedthat P2(x) ∼ λ2x(log x)−2. We prove that, for suf�ciently large x and for asuitable constant k, P2(x)> x exp{−k(log x)1/2}.

### Su alcune successioni di interi

"... 1 A positive integer m is a practical number if every positive integer n < m is a sum of distinct positive divisors of m. Definition 2 Let P (x) be the counting function of practical numbers. Definition 3 Let P2(x) be the function counting practical numbers m x such that m + 2 is also a practica ..."

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1 A positive integer m is a practical number if every positive integer n < m is a sum of distinct positive divisors of m. Definition 2 Let P (x) be the counting function of practical numbers. Definition 3 Let P2(x) be the function counting practical numbers m x such that m + 2 is also a practical number. Theorem 1 (Stewart, 1954) A positive integer m 2; m = q11 q22 qkk; with primes q1 < q2 < < qk and integers i 1; is practical if and only if q1 = 2 and, for i = 2; 3; : : : ; k; qi q11 q

### Some Information-Theoretic Computations Related to the Distribution of Prime Numbers

, 2007

"... We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all prim ..."

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We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finite-n bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.