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Search: Posts Made By: Dr Sardonicus
Forum: Number Theory Discussion Group 2021-05-08, 13:37
Replies: 6
Views: 209
Posted By Dr Sardonicus
I forgot to point out that the thing is also...

I forgot to point out that the thing is also trivial when n = 1, y^2 = 0 (mod 1).

What I proved was that your given hypotheses imply that

2*n - 1 divides 4*y^2 + 1 if n > 1

2*|n| + 1 divides...
Forum: Soap Box 2021-05-08, 02:45
Replies: 47
Views: 5,588
Posted By Dr Sardonicus
He gave the gun to the kid and ran. The...

He gave the gun to the kid and ran. The following is from the Chicago Tribune, which I believe qualifies as "mainstream media."

Prosecutors detail the night police shot Adam Toledo, 13, during...
Forum: Number Theory Discussion Group 2021-05-08, 02:18
Replies: 6
Views: 209
Posted By Dr Sardonicus
(y^2 - (n^2 - n))/(2*n - 1) is an integer when ...

(y^2 - (n^2 - n))/(2*n - 1) is an integer when

n^2 - n == y^2 (mod 2*n-1)

4*n^2 - 4*n == 4*y^2 (mod 2*n - 1) †

(2*n - 1)^2 == 4*y^2 +1 (mod 2*n - 1)

4*y^2 + 1 == 0 (mod 2*n - 1); that is,
Forum: Soap Box 2021-05-07, 15:43
Replies: 338
Views: 38,126
Posted By Dr Sardonicus
Meanwhile, back in the real world... UK...

Meanwhile, back in the real world...

UK ships prepare to leave Jersey after dispute over fishing rights (https://www.bbc.com/news/uk-57016222)
Forum: Soap Box 2021-05-07, 13:29
Replies: 182
Views: 29,716
Posted By Dr Sardonicus
Hmm, sounds like the Ukrainian authorities are...

Hmm, sounds like the Ukrainian authorities are engaging in what we in the good ol' USA call a "shakedown."

I imagine the officials responsible could simply claim they "mistook" valid UK excise...
Forum: Lounge 2021-05-07, 13:12
Replies: 221
Views: 35,528
Posted By Dr Sardonicus
Flock of giant California condors trashes woman's...

Flock of giant California condors trashes woman's home (https://apnews.com/article/california-condors-10c57295075a7ca1d33a88d759bfcc52)
Forum: Soap Box 2021-05-07, 12:06
Replies: 338
Views: 38,126
Posted By Dr Sardonicus
Yes, even in the good ol' USA it was called the...

Yes, even in the good ol' USA it was called the "Norman conquest" in history classes when I went to school.

Even so, the context of your "We still haven't forgotten 1066 and all that" clearly...
Forum: Lounge 2021-05-07, 01:40
Replies: 67
Views: 12,705
Posted By Dr Sardonicus
You might want to look at The 2021 Periodical...

You might want to look at The 2021 Periodical Cicada Emergence (Brood X) (https://cicadas.uconn.edu/) (which also has links to "stragglers"). The striking thing about periodical cicadas is the...
Forum: Lounge 2021-05-06, 22:26
Replies: 277
Views: 36,630
Posted By Dr Sardonicus
An AP news story from Maine, possibly related to...

An AP news story from Maine, possibly related to the subject of this thread, may be found here (https://apnews.com/article/maine-government-and-politics-c05ec4fc0b20368537d9657b9128234a).
Forum: Soap Box 2021-05-06, 20:29
Replies: 338
Views: 38,126
Posted By Dr Sardonicus
You checked here...

You checked here (https://www.mersenneforum.org/showpost.php?p=577800&postcount=316), no doubt.
Forum: Soap Box 2021-05-06, 13:43
Replies: 338
Views: 38,126
Posted By Dr Sardonicus
:missingteeth: :missingteeth: :missingteeth:...

:missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth: :missingteeth:...
Forum: Lounge 2021-05-06, 12:45
Replies: 221
Views: 35,528
Posted By Dr Sardonicus
Rare hummingbird's stop in LaBagh Woods is a...

Rare hummingbird's stop in LaBagh Woods is a once-in-a-lifetime sight for Chicago birders...
Forum: Lounge 2021-05-05, 16:21
Replies: 67
Views: 12,705
Posted By Dr Sardonicus
Here's a "topper." Some periodical cicadas spend...

Here's a "topper." Some periodical cicadas spend 13 years underground. And 13 is another prime number! Thus,the periods are relatively prime, so a given pair of broods of 17-year and 13-year...
Forum: Puzzles 2021-05-05, 12:06
Replies: 16
Views: 774
Posted By Dr Sardonicus
Should there be a A_0 and A_1 that satisfies this...

Should there be a A_0 and A_1 that satisfies this for every k?[/quote]Yes. Using the condition that the pk are distinct, the existence of simultaneous solutions to all the congruences is guaranteed...
Forum: Lounge 2021-05-05, 12:00
Replies: 67
Views: 12,705
Posted By Dr Sardonicus
This may affect storm5510 since he lives in...

This may affect storm5510 since he lives in Indiana...

Nature at its craziest: Trillions of cicadas about to emerge...
Forum: Lounge 2021-05-04, 20:52
Replies: 67
Views: 12,705
Posted By Dr Sardonicus
When xilman talks, the EU listens... Food of...

When xilman talks, the EU listens...

Food of the future? EU nations put mealworms on the menu (https://apnews.com/article/europe-worms-science-business-oddities-c12bf770addd95eed248a0deee5e5d95)
Forum: Lounge 2021-05-04, 19:19
Replies: 40
Views: 2,685
Posted By Dr Sardonicus
Long minutes googling turned up the explanation...

Long minutes googling turned up the explanation that - it's tradition!

Part of the tradition is, the names of the months were generally written out. The Declaration of Independence has the date,...
Forum: Puzzles 2021-05-04, 12:50
Replies: 16
Views: 774
Posted By Dr Sardonicus
The May 2021 Challenge seems quite similar to...

The May 2021 Challenge seems quite similar to finding Riesel or Sierpinski numbers by means of "covering sets" of congruences.

The misstatement of the subscripts is the kind of oopsadaisy a number...
Forum: Lounge 2021-05-04, 02:12
Replies: 221
Views: 35,528
Posted By Dr Sardonicus
Traveling owl: Man drives hours with bird in...

Traveling owl: Man drives hours with bird in truck's grille (https://apnews.com/article/birds-owls-oddities-ec6653cef99156ee11710bb15f8da832)
Forum: Science & Technology 2021-05-03, 13:56
Replies: 205
Views: 10,969
Posted By Dr Sardonicus
One quotation, attributed to H. L. Mencken: "No...

One quotation, attributed to H. L. Mencken: "No one in the world, so far as I know - and I have searched the records for years, and employed agents to help - has ever lost money by underestimating...
Forum: Soap Box 2021-05-03, 12:45
Replies: 117
Views: 28,701
Posted By Dr Sardonicus
The cause that led Martin Luther King, Jr. to...

The cause that led Martin Luther King, Jr. to Memphis, where he was murdered on April 4, 1968, as described in this post (https://www.mersenneforum.org/showpost.php?p=512651&postcount=64), was that...
Forum: Lounge 2021-04-30, 12:35
Replies: 1,363
Views: 118,943
Posted By Dr Sardonicus
Two influencers ordered off Bali after fake mask...

Two influencers ordered off Bali after fake mask video prank (https://apnews.com/article/asia-pacific-bali-health-coronavirus-arts-and-entertainment-4ec6747d25b03487956e02651aefd7e4)
Forum: Homework Help 2021-04-29, 15:35
Replies: 14
Views: 613
Posted By Dr Sardonicus
The Archimedean property is that if x and y are...

The Archimedean property is that if x and y are in an ordered field, x > 0 and y > 0, there is a positive integer n such that n*x > y.

Assuming the ordered field is the field of rational numbers...
Forum: Data 2021-04-29, 11:14
Replies: 565
Views: 88,887
Posted By Dr Sardonicus
A 61 decimal digit factor winkled out. Amazing! ...

A 61 decimal digit factor winkled out. Amazing!

Whatever the problem was with the PRP cofactor residue, it's been fixed. Two tests show a residue of PRP_PRP_PRP_PRP_

And, as previously...
Forum: Data 2021-04-29, 00:11
Replies: 565
Views: 88,887
Posted By Dr Sardonicus
I'm curious. Usually when a cofactor tests as a...

I'm curious. Usually when a cofactor tests as a PRP, the residue is listed as

PRP_PRP_PRP_PRP_

But here, although the history says the remaining cofactor is a probable prime, the PRP cofactor...
Showing results 1 to 25 of 1000

 
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