By Paulo Ribenboim

This feature of expository essays via Paulo Ribenboim may be of curiosity to mathematicians from all walks. Ribenboim, a hugely praised writer of a number of well known titles, writes each one essay in a mild and funny language with out secrets and techniques, making them completely available to all people with an curiosity in numbers. This new assortment contains essays on Fibonacci numbers, leading numbers, Bernoulli numbers, and historic displays of the most difficulties concerning basic quantity concept, resembling Kummers paintings on Fermat's final theorem.

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**Additional resources for My Numbers, My Friends: Popular Lectures on Number Theory**

Ak− 1 are optimistic integers that won't be coprime). A 2- robust quantity is just known as a strong quantity. specifically, strong numbers are these of the shape a 20 a 31, with a zero , a 1 ≥ 1. We notice that 1 is a robust quantity. I shall denote by way of Wk the set of k-powerful numbers. the most difficulties approximately robust numbers are of the subsequent types: 1. Distribution of strong numbers. 2. Additive difficulties. three. distinction difficulties. A. Distribution of strong numbers the purpose is to estimate the variety of parts within the set Wk( x) = {n ∈ Wk | 1 ≤ n ≤ x}, (1) the place x ≥ 1, okay ≥ 2. Already in 1935, Erdös and Szekeres gave the 1st end result approximately W 2( x): ζ( three ) # W 2 2( x) = x 1 / 2 + O( x 1 / three) as x → ∞, (2) ζ(3) the place ζ( s) is the Riemann zeta functionality; see additionally Bateman (1954) and Golomb (1970). to explain the newer effects, I introduce the zeta functionality linked to the series of k-powerful numbers. allow 1 if n is k-powerful, jk( n) = zero differently. The sequence ∞ jk( n) n=1 and defines a ns is convergent for Re( s) > 1 okay functionality Fk( s). This functionality admits the next Euler product 1 strong numbers 231 illustration 1 1 F pks okay( s) = 1 + = 1 + , (3) 1 − 1 p( okay− 1) s( playstation − 1) p playstation p that is legitimate for Re( s) > 1 . ok With famous equipment, Ivić and Shiu confirmed in 1982: 1 1 1 (1. 1) # Wk( x) = γ zero ,kx okay + γ 1 ,kx k+1 + · · · + γk− 1 ,kx 2 ok− 1 + ∆ ok( x), the place γi,k is the residue at 1 of Fk( s) . okay+ i s Explicitly, Φ okay( 1 ) γ okay+ i i,k = Ck+ i,k , (4) ζ( 2 k+2 ) ok+ i the place 2 okay− 1 j Ck+ i,k = ζ , (5) ok + i j= ok j= okay+ i Φ2( s) = 1, and if ok > 2, then Φ okay( s) has a Dirichlet sequence with abscissa of absolute convergence 1 , and ∆ 2 k+3 okay( x) is the mistake time period. Erdös and Szekeres had already thought of this mistake time period and confirmed that 1 ∆ ok( x) = O( x k+1 ) as x → ∞. (6) larger estimates of the mistake have in view that been acquired. allow ρk = inf {ρ > zero | ∆ ok( x) = O( xρ) }. Bateman and Grosswald confirmed in 1958 that ρ 2 ≤ 1 and ρ . 6 three ≤ 7 forty six Sharper effects are as a result of Ivić and Shiu: 1 7 ρ 2 ≤ zero . 128 < , ρ three ≤ zero . 128 < , ρ four ≤ zero . 1189 , 6 forty six ρ five ≤ 1 , ρ 6 ≤ 1 , ρ 7 ≤ 1 , and so on. 10 12 14 I refer additionally to the paintings of Krätzel (1972) in this topic. it truly is conjectured that, for each okay ≥ three, 1 ∆ okay( x) = O( x 2 okay ) for x → ∞. (7) 232 nine. Powerless dealing with Powers extra particularly, taking okay = 2: ζ( three ) 1 ζ( 2 ) 1 # W 2 three 2( x) = x 2 + x three + ∆2( x) , (8) ζ(3) ζ(2) 1 with ∆2( x) = O( x 6 ), as x → ∞. B. Additive difficulties If h ≥ 2, ok ≥ 2, I shall use the subsequent notation: h hWk = { ni | each one ni ∈ Wk ∪ { zero }}, i=1 hWk( x) = {n ∈ hWk | n ≤ x} (for x ≥ 1). The additive difficulties trouble the comparability of the units hWk with the set of traditional numbers, the distribution of the units hWk, and comparable questions. The distribution of 2 W 2 used to be taken care of by way of Erdös in 1975: (1. 2) x # 2 W 2( x) = o (as x → ∞) , the place zero < α < 1 . (log x) α 2 specifically, # 2 W 2( x) = o( x), so there exist infinitely many usual numbers which aren't the sum of 2 robust numbers.