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.

This added restriction allowed the

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 <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] > I wonder if the largest pair list at A002072 should

These values are indeed the largest known

> 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.

(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

For p = 229 we have 5.2*sqrt(p)-7.7 = 71,

> found this pair (S, S+1):

>

> S = 15487655655079919751646464

so I expect N about exp(71), which is much

larger than your N about exp(58).

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,

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]