A search for very low Nash weights

I have previously used a convenient method for finding numbers with low Nash weight. This consists of counting the number of times that the Robinson numbers k.2ⁿ + 1 for n between 1 and 1000 do not succumb to trial division by the primes less than 2000. If this count exceeds 10 then the value of k is discarded, otherwise I calculate the Nash weight. I first extended this idea for all odd k less than 10⁷. This provides the following data (where NW(k) is used as shorthand for the Nash weight).

Alternatively, we can count the under / over 100 split for each intermediate weight, as follows.

The first percentage above is of NW < 100 against total for that weighting, (column 3 against column 2) and the second percentage is of the particular weighting (column 2 ) against the total number of k considered.

From the above data, it can be seen that the return in terms of number of values with a Nash weight less than 100 peaks at a weighting of 8 and tails off for 9 and 10. Since these take extra time to locate and account for more than half of the total, any additional searches will in future restrict the intermediate weighting further in order to speed things up and reduce the volume of data to be considered. Since in the current range, the lowest Nash weights found for weightings 9 and 10 are 67 and 73 respectively, it is unlikely that a Nash weight of less than 50 will be overlooked by this restriction.

There were 69 values of k found with a Nash weight of zero. These have all been verified as Sierpinski numbers. Apart from these, the lowest Nash weights found in the range being considered were as follows.

Nash weights as low as this are striking and indicate the possibility of Sierpinski numbers with infinite covering sets, a concept about which little is known at present. However, each of the above was checked for n £ 10000, and the numbers marked with an asterisk have associated Keller primes for n = 4, 2193 and 28 respectively. A very low Nash weight, by itself, is therefore no absolute guarantee of the Sierpinski property, and suggests that additional forces are at work.

Consider k = 3203597. A quick check reveals that the only values of n up to 8000 for which k.2ⁿ +1 is not divisible by a prime less than 2000 are as follows.

The factors above were found relatively quickly using either a straight sieve or Pollard's (p- 1)-method. This list can be extended by using Gallot's proth.exe program with the appropriate settings. Including the three above, there are only 72 values of n £ 100000 that require to be fully tested to determine their nature.

The latest version of psieve.exe includes parameters -ya and -zb which redefine the sieve range to start at a and to run across b consecutive values. The standard Nash weight is obtained using a = 100000 and b = 10000. If we take a = 1 and b = 100000, we get a more sensitive measure of the effectiveness of the sieve.

Extending the search to 2*10⁷ but with a limit of 8 on the intermediate weighting provides the following cumulative results.

Of the values with intermediate weighting of 0, one, that is, k = 13965257, has a non-zero Nash weight of 34 and is not a Sierpinski number. Other numbers of very low Nash weight include the following.

The marked numbers have Keller primes for n = 947, 7263, 3 and 93 respectively with searches limited to n £ 10000.

Consider k = 19989199. The only values of n up to 8000 for which k.2ⁿ +1 is not divisible by a prime less than 2000 are the following.

Including the three above, there are only 65 values of n £ 100000 that require to be fully tested by proth.exe to determine their nature. With the different sieve limits, we obtain the following values.

Extending the search to 5*10⁷ but with a limit of 6 on the intermediate weighting provides the following cumulative results.

Another value with intermediate weighting of 0, that is, k = 47794969, has a non-zero Nash weight of 37 and is not a Sierpinski number. Other numbers of very low Nash weight include the following.

The marked numbers have Keller primes for n = 455, 475, 409, 724 and 67 respectively with searches limited to n £ 10000. The value k = 29607287 is remarkable since as well as providing a prime for n = 455, it also provides a prime for n = 9383. Similarly, k = 43936787 provides a prime for n = 355 as well as n = 67.

Consider k = 46082329. The only values of n up to 8000 for which k.2ⁿ +1 is not divisible by a prime less than 2000 are the following.

Including the seven above, there are only 51 values of n £ 100000 that require to be fully tested by proth.exe to determine their nature. With the different sieve limits, we obtain the following values.

Extending the search all the way to 10⁸ with a limit of 5 on the intermediate weighting provides the following cumulative results.

Another 4 values were found with an intermediate weighting of zero but a non-zero Nash weight. Other numbers of very low Nash weight include the following.

The marked numbers have Keller primes for n = 49, 57, 198 and 770 respectively with searches limited to n £ 10000.

Consider k = 57159653. The only values of n up to 8000 for which k.2ⁿ +1 is not divisible by a prime less than 2000 are the following.

Including the three above, there are only 50 values of n £ 100000 that require to be fully tested by proth.exe to determine their nature. With the different sieve limits, we obtain the following values.

Extending the search to 2*10⁸ with a limit of 4 on the intermediate weighting provides the following cumulative results.

Another 3 values were found with an intermediate weighting of zero but a non-zero Nash weight. Other numbers of very low Nash weight include the following.

The marked number has a Keller prime for n = 997. All the above values of k were searched to a limit of n £ 10000.

Consider k = 166851887. The only values of n up to 8000 for which k.2ⁿ +1 is not divisible by a prime less than 2000 are the following.

Including the two above, there are only 28 values of n £ 100000 that require to be fully tested by proth.exe to determine their nature. With the different sieve limits, we obtain the following values.

Extending the search to 5*10⁸ with a limit of 2 on the intermediate weighting provides the following cumulative results.

There were 16 additional values with an intermediate weighting of zero but a non-zero Nash weight. Although no value bettered the Nash weight of 9 given above, the following have very low Nash weight.

The marked numbers have Keller primes for n = 1977 and 4076 respectively with searches limited to n £ 10000.

The total number of k values found with Nash weight less than 20 is 29, all of which are listed above. For completeness, the next table gives numbers of k found with Nash weights in the given ranges.

Note that the counts drop off because of the increasing restrictions applied to the search.

URL : www.glasgowg43.freeserve.co.uk/nash2.htm