## RE reciprocal consecutive primes

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----- Original Message -----

From: "Milton Brown" <miltbrown@...>

Actually, it doesn't seem that hard:

1/p < 1/q + 1/r

q>=p+2

r>=q+2>=p+4

It must be shown that 1/p < 1/(p+2) + 1/(p+4)

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FALSE. If r>=p+4, then 1/r <= 1/(p+4). Considering the case where 1/r = 1/(p+4) and 1/q =

1/(p+2) is a best case study, not a worst case one.

Jose Brox

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----- Original Message -----

From: "Jan van Oort" <glorifier@...>

p^2 > mn

which is true iff

( m < p AND n < p ) OR ( m = p + a AND n = p - b AND b > a ) ( condition i )

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Jose:

That's not true.

Consider b = a-1, then we have

m*n = (p+a)(p-b) = (p+a)(p-a+1) = p^2 +p-a(a+1)

If a(a+1) > p, then m*n < p^2, with b < a, m=p+a, n = p-b. You can get similar results

with different b < a.

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Jan:

now condition ( i ) only holds iff Bertrand's postulate holds

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Jose:

Can you explain why? I can't see it at the moment.

Jose Brox

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