Number Composite Astronomical M281414479

[ewmayer]

Originally Posted by CRGreathouse

Yes. Actually I was surprised at how hard it was to find good chains (let alone optimal chains).

But unless my skim of same is egregiously off, the chains-page you supplied makes clear that things are very sum-specific (exponent-specific in a TF context), and there does not appear to be a fast deterministic algorithm for finding the optimal chain, or even on appreciably superior to that given by the standard LR binary modpow approach. If we could do that cheaply on-the-fly it would certainly be worth considering as an enhancement to our TF codes.

[CRGreathouse]

Originally Posted by ewmayer

But unless my skim of same is egregiously off, the chains-page you supplied makes clear that things are very sum-specific (exponent-specific in a TF context), and there does not appear to be a fast deterministic algorithm for finding the optimal chain, or even on appreciably superior to that given by the standard LR binary modpow approach. If we could do that cheaply on-the-fly it would certainly be worth considering as an enhancement to our TF codes.

Indeed -- provably so, on P = NP. So in this case theory matches practice pretty well.

[PaulineEinstein]

Thanks Raman
Your contribution has been very constructive for me. I move on in my studies. Not bother again.
Ha. I am Colombian
I Sorry!
bye.

[Raman]

Originally Posted by Raman

Consider with double Wagstaff prime number exponents, for an example!
(2⁷+1)/3 = 43 is prime number.
(2⁴³+1)/3 = 2932031007403 is prime number.
(2^{2932031007403}+1)/3 has got no small prime factors below what?
What will be your bet to that type of thing?

Double Wagstaff numbers:

Starting point 3:

(23+1)/3 = 3 is prime number.
(23+1)/3 = 3 is prime number.
...
cycle.

Starting point 5:

(25+1)/3 = 11 is prime number.
(211+1)/3 = 683 is prime number.
(2683+1)/3 = 1676083 × 26955961001 × 296084343545863760516699753733387652635366098889116410731661924253563729059085336779932810899819313612925255002666691226800507277398580985624625950496168983999760414855301693388419156899841.

Starting point 7:

(27+1)/3 = 43 is prime number.
(243+1)/3 = 2932031007403 is prime number.
(22932031007403+1)/3 is unknown number...

Double Mersenne numbers:

Starting point 5:

25-1 = 31 is prime number.
231-1 = 2147483647 is prime number.
22147483647-1  has the factors: 295257526626031, 87054709261955177, 242557615644693265201, 178021379228511215367151.

Starting point 2, 3, 7:

22-1 = 3 is prime number.
23-1 = 7 is prime number.
27-1 = 127 is prime number.
2127-1 = 170141183460469231731687303715884105727 is prime number.
2170141183460469231731687303715884105727-1 is unknown number...

(May be that some one can prove that 2^{170141183460469231731687303715884105727}-1
is being immediately composite if any number of that number is not being of the following
form as like (2 × 170141183460469231731687303715884105727 × k) + 1 )

(Non-algebraic factors of 2^{170141183460469231731687303715884105727}-1 will be
always of the following form as like (2 × 170141183460469231731687303715884105727 × k) + 1).

This mersenne forum thread post out.
1234 posts out.
Counting out.
Existing out.
Exactly out.
Very quite variably - about around - round ground - poultry variety - out.

Originally Posted by Raman

Remember ≠ forget!
Or not showing out thoughts and strikings to you all.
Every thing every where every body every one every time - all though all together all beit all most all right.
Else thing else where else body else one else time - at though at together at beit at most at right.
Last past time period.

Kept away - stay off - away up out off down my own
nfsnet.org web site page showing out as like
some breast cancer treatment type of thing?
that ever which ever a way a way ever - by using be being

Double quotes, not single quotes not needed up for url tags
but that it is being always required up for colour tags?

Color, not colour?
Gray, not grey?



Plug sockets shapes, sizes, colours, strokes and scripts in different countries.

Up.
SNFS Biquadratic = SNFS Quartic?
Is SNFS quartic being a much further more better term to use up rather than instead of SNFS biquadratic?
Most further more.

Always preparing up with drafts in ready made all tags!

[CRGreathouse]

Originally Posted by Raman

SNFS Biquadratic = SNFS Quartic?
Is SNFS quartic being a much further more better term to use up rather than instead of SNFS biquadratic?

People use the term differently. The old-fashioned meaning of biquadrate/biquadratic is a fourth-degree polynomial, but it's also used with the narrower meaning of a fourth-degree polynomial with odd exponents having coefficient 0. So x^4 - 17x^2 + 2 is a biquadrate under either definition but x^4 + x is only in the former (people using the latter meaning would call this a quartic).

I recommend using "quartic" when you mean (only) a fourth-degree polynomial as it is unambiguous, and if you want to use "biquadratic" in the second sense you should define it in the text so that readers aren't confused.

[NeillC]

Originally Posted by CRGreathouse

Using this nifty tool (due to Neill Clift and Achim Flammenkamp) you can see that this can be improved to 34 modular multiplications, the 28 squarings above and just 6 other multiplications. I haven't found an actual addition chain of this length, though; anyone want to try?

Just saw this sorry.
It doesn't follow that a number with has 28 doublings. It may have less doublings and more general multiplies.

An addition chain of length 34 follows. I didn't search for one with the most doublings and haven't checked this one.

1 2 4 6 12 24 48 96 192 384 768 1536 3072 6144 12288 24576 49152 98304 196608 202752 202753 399361 798722 1001475 2002950 4005900 8011800 16023600 17025075 33048675 66097350 132194700 132194702 264389404 281414479