## Known pairs vs OEIS A002072

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•   I wonder if the largest pair list at A002072 should include some statement that certain values are smallest known rather than smallest , as the latter
Message 1 of 4 , Mar 7, 2012

I wonder if the largest pair list at A002072 should
include some statement that certain values are
"smallest known" rather than "smallest", as the
latter assumes the values can be confirmed in
some rigorous manner.

I haven't done a rigorous Lehmer-method in yonks,
I can't remember at what stage I started applying
period bailout, somewhere around p = 97.

p = 97 is, as far as I know, the greatest prime
for which the results are rigorous, this result
being quite recent, ie Luca & Najman (2010).

They used a very non-Lehmer method involving
computing regulators.

It's a very expensive method, thousands of times
slower than my modified Lehmer (period bailout),
but probably hundreds of times faster than a
"rigorous Lehmer" would be.

The results I computed up to p=173 are
probably correct, although the monster at p=157
casts reasonable doubt over all subsequent
results.  By this stage I was not just applying
period bailout, but also a bailout on the
number of prime divisors of D.

computations to be completed in days rather
than months, but the probability that I missed
bigger monsters is accordingly increased.

Having said that, and assuming both A002072 and
column k=1 of your chain table are "smallest
known", I do have a new entry for p = 227, having
found this pair (S, S+1):

S = 15487655655079919751646464

This was found via a PTE match at
Q = 395684061 {0, 3, 3}, {1, 1, 4}

Thus S = Q(Q+3)^2 / 4

Jim White
Canberra

________________________________
From: Andrey Kulsha <andrey_601@...>
To: Jim White <mathimagics@...>
Sent: Wednesday, 7 March 2012, 4:29
Subject: Re: [PrimeNumbers] Re: 13-chains of consecutive smooth numbers

﻿
Any chance of a text export?
Here we are: http://www.primefan.ru/stuff/math/maxs.txt

Chain length from 6 to
16, N up to 10^13.

Best regards,

Andrey

[Non-text portions of this message have been removed]
• I m sorry, I said 227, I really meant 229! ________________________________ From: Jim White  Subject: [PrimeNumbers] Known pairs vs
Message 2 of 4 , Mar 7, 2012
I'm sorry, I said 227, I really meant 229!

________________________________
From: Jim White <mathimagics@...>
Subject: [PrimeNumbers] Known pairs vs OEIS A002072

Having said that, and assuming both A002072 and
column k=1 of your chain table are "smallest
known", I do have a new entry for p = 227, having
found this pair (S, S+1):

S = 15487655655079919751646464

This was found via a PTE match at
Q = 395684061 {0, 3, 3}, {1, 1, 4}

Thus S = Q(Q+3)^2 / 4

Jim White
Canberra

________________________________
From: Andrey Kulsha <andrey_601@...>
To: Jim White <mathimagics@...>
Sent: Wednesday, 7 March 2012, 4:29
Subject: Re: [PrimeNumbers] Re: 13-chains of consecutive smooth numbers

﻿
Any chance of a text export?
Here we are: http://www.primefan.ru/stuff/math/maxs.txt

Chain length from 6 to
16, N up to 10^13.

Best regards,

Andrey

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• ... These values are indeed the largest known (there may be larger ones). Doubt is not about smallest but about for all i m . ... For p = 229 we have
Message 3 of 4 , Mar 10, 2012
> I wonder if the largest pair list at A002072 should
> include some statement that certain values are
> "smallest known" rather than "smallest", as the
> latter assumes the values can be confirmed in
> some rigorous manner.

These values are indeed the largest known
(there may be larger ones). Doubt is not about
"smallest" but about "for all i>m".

> I do have a new entry for p = 229, having
> found this pair (S, S+1):
>
> S = 15487655655079919751646464

For p = 229 we have 5.2*sqrt(p)-7.7 = 71,
so I expect N about exp(71), which is much

Best regards,

Andrey
• This 181-smooth pair popped up in a random search:   S = 90672220863645734556839376   Factors   S   = 2^4, 3^3, 11, 23, 29, 31, 37^2, 73, 97,
Message 4 of 4 , Mar 12, 2012
This 181-smooth pair popped up in a random search:

S = 90672220863645734556839376

Factors
S   = 2^4, 3^3, 11, 23, 29, 31, 37^2, 73, 97,
139^2, 163, 167, 181
S-1 = 5^4, 13^3, 17, 19, 43, 47, 67^2, 101^2,
113^2, 173

[Non-text portions of this message have been removed]
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