Problems & Puzzles: Puzzles

 Puzzle 760. A087711 Jahangeer Kholdi & Farideh Firoozbakht sent the following nice puzzle.   For positive integers n, we define a(n), b(n) & c(n) as follows. a(n) = Smallest number m such that phi(m+n) + sigma(m-n) = 2m. b(n) = Smallest number m such that phi(m-n) + sigma(m+n) = 2m. c(n) = Smallest number m such that phi(m+n) + sigma(m-n) = phi(m-n) + sigma(m+n). Q. Prove that for each positive integers n, a(n) = b(n) = c(n) = A087711(n) or find a counterexample.

Contribution came from Fred Schneider
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Fred wrote:

Seems like c(n) is not necessary. If a(n) = b(n), c(n) must be satisfied, as they are the same formulas.

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Jarek Biszczuk wrote on Oct 31, 2014:

I present the results of what I found to puzzle 760, can inspire someone to more complete results.

Let delta(x,y)=sigma(x)+phi(y)-x-y

Properties delta(x,y):
For p in P sigma(p)=p+1, phi(p)=p-1, sigma(p^2)=1+p+p^2, phi(p^2)=p^2-p
For p,q in P delta(p,q)=0
if delta(x,y)=0 or delta(x,y)=delta(y,x) then both x,y are prime or composite
delta(x,x)>0 iff. x is composite
delta(x,x)=0 iff. x is prime

Let
alpha(n)=min{m: delta(m,m+2n)=0}
beta(n)=min{m: delta(m+2n,m)=0}
gamma(n)=min{m: delta(m+2n,m)=delta(m,m+2n)}
A020483(n)=min{m: delta(m,m+2n)=0 and m is prime}
equivalent definitions sequence A020483:
1) A020483(n)=min{m: delta(m+2n,m)=0 and m is prime}
2) A020483(n)=min{m: delta(m+2n,m)=delta(m,m+2n) and m is prime}
A020483(n) exists for each n, if the conjecture is true that every even number can be
written as the difference of two primes.
max{A020483(n): n<5e11}=3307 see http://oeis.org/A020483

In Puzzle760:
a(n)=alpha(n)+n
b(n)=beta(n)+n
c(n)=gamma(n)+n
A087711(n)=A020483(n)+n

Puzzle 760:
Find composite m<A020483(n) and (
delta(m,m+2n)=0 or
delta(m+2n,m)=0 or
delta(m,m+2n)=delta(m+2n,m))
or prove that m not exists.

Let 4<=h<3307 and h composite,
H>7650^2 and H composite
then
-2500 <= phi(h)-h <= -2
3 <= sigma(h)-h <= 7650
phi(H)-H <= -7669
sigma(H)-H >= 7670
and delta(H,h) > 0, delta(h,H) < 0.
Numarical searching for n<7650^2: no exists composite m.
Summary no exist composite m for puzzle 760 with n<5e11.

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