## Primes >2

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• Prime numbers greater than two are those numbers, expressed in base two, which contain a one in the rightmost position and and odd number of ones, or
Message 1 of 6 , Jul 6, 2011
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Prime numbers greater than two are those numbers, expressed in base two,
which contain a one in the rightmost position and and odd number of
ones, or (excluding leading zeroes) an odd number of zeroes, if there
are any.

Any base two number greater than two, with a zero in the rightmost
position is naturally a multiple of two, and hence not prime.

Any base two number with an even number of zeroes is naturally a
multiple of two smaller numbers, and hence not prime.

Dana E. Edgecomb
• ... So 73 = 64 + 8 + 1 = (1001001)_2 is the product of ???
Message 2 of 6 , Jul 6, 2011
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> Any base two number with an even number of zeroes is naturally a

So 73 = 64 + 8 + 1 = (1001001)_2 is the product of ???
• ... Not so. print(binary(97)) [1, 1, 0, 0, 0, 0, 1] David
Message 3 of 6 , Jul 6, 2011
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Dana Edgecomb <dedgecomb@...> wrote:

> Any base two number with an even number of zeroes is naturally
> a multiple of two smaller numbers, and hence not prime.

Not so.

print(binary(97))
[1, 1, 0, 0, 0, 0, 1]

David
• ... How about 2^43112609-1? It has zero -- an even number -- of 0 s and an odd number of 1 s, in base 2. 7 = (111)_2 also fits, Paul
Message 4 of 6 , Jul 7, 2011
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--- In primenumbers@yahoogroups.com, Dana Edgecomb <dedgecomb@...> wrote:
>
>
> Prime numbers greater than two are those numbers, expressed in base two,
> which contain a one in the rightmost position and and odd number of
> ones, or (excluding leading zeroes) an odd number of zeroes, if there
> are any.
>
>
> Any base two number greater than two, with a zero in the rightmost
> position is naturally a multiple of two, and hence not prime.
>
>
> Any base two number with an even number of zeroes is naturally a
> multiple of two smaller numbers, and hence not prime.
>

How about 2^43112609-1? It has zero -- an even number -- of 0's and an odd number of 1's, in base 2.

7 = (111)_2 also fits,

Paul
• ... Although Mr. Edgecomb is incorrect in this particular, the general principle that we can eyeball some numbers and quickly exclude them from primeness,
Message 5 of 6 , Jul 8, 2011
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--- In primenumbers@yahoogroups.com, Dana Edgecomb <dedgecomb@...> wrote:
>
>
> Prime numbers greater than two are those numbers, expressed in base two,
> which contain a one in the rightmost position and and odd number of
> ones, or (excluding leading zeroes) an odd number of zeroes, if there
> are any.
>
>
> Any base two number greater than two, with a zero in the rightmost
> position is naturally a multiple of two, and hence not prime.
>
>
> Any base two number with an even number of zeroes is naturally a
> multiple of two smaller numbers, and hence not prime.
>
> Dana E. Edgecomb

Although Mr. Edgecomb is incorrect in this particular, the general principle that we can eyeball some numbers and quickly exclude them from primeness, holds.

For instance, take any arrangement of digits yielding a number greater than one. If that digit arrangement occurs in a number more than once (and not overlapping), and the rest of the digits are zeros, we can know that the number is not prime.

Specifically, the number 101000101 is not prime because it has the digit arrangement 101 more than once, and the rest are zeros. This holds whether the number above was in binary notation or (say) decimal notation.

The mathematical reason for this should be clear. :)

Every composite number has its weakness. There will be some base in which to express such a number, which eyeballing will reveal to be composite. Of course, the number must be humanly sized. :)
• ... This is true, in a unhelpful way. Suppose that N is divisible by b, where N b 1. Then N ends in 0, when written in base b. As remarked, this not
Message 6 of 6 , Jul 8, 2011
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