Invertebrates and Potables (I & P) 51

@morphyms1817 - Bio 51 Lab Invertebrate Key Flash cards; Used to learn the many kinds of Invertebrate species. Maybe some of that Area 51 stuff. - :]

@Noflaps said... Kung Fu. Why can't they make such excellent television nowadays? How I miss it.

I was surprised when I found out that walking across rice paper without disturbing it, and snatching pebbles from hands, did not lead to a higher starting salary, no matter how much I practiced at it.

That's very good 'Grass Hopper' You can see all the fight scenes from Kung Fu, on Utube. Stayed up all night once watching them. I loved that show, and the movies too. - :]

@Noflaps said in #18:

Furthermore, now that the word "hippopotamus" has appeared in the thread, I am quite free to sing the praises of the lovely Hippo.

One does not choose to play the Hippo. The Hippo chooses one to play it.

Its pacific symmetry calls out to few. Most look at the Hippopotamus Opening and snicker. Rudely! They sometimes even elbow each other and smirk out "hey, look at THIS silly opening!"

But soon their shallow smirks begin to fade, as they dimly begin to sense -- from the base of their reptilian brain -- that the Hippo is far mightier -- far better -- than it at first appears.

Aaah, the ever aquatic river horse + the pacific + invertebrates = the formidable multicoloured vibrances called sea slugs.

Quite toxic, these entities, poisonous even, but a sheer delight to behold. We should have seen this coming.

@Noflaps said in #20:

Oh, you're plenty sharp yourself, @morphyms1817 . Sharpness comes in many varieties -- for many areas of emphasis.

As for me, I became skilled at numbers from the time I began counting sheep in order to slumber. And since I love to sleep, it blossomed into a hobby.

There are quite a few math mavens skulking around the forum, actually. As Kwai Chang Caine once said (more or less): I know them ... by the way .... they move.

Having written that, it makes me realize for the first time that William Shatner could also have made an excellent Kwai Chang Caine! After all .. Captain Kirk ... often spoke with ... the same ... vocal rhythm.

Kung Fu. Why can't they make such excellent television nowadays? How I miss it.

I was surprised when I found out that walking across rice paper without disturbing it, and snatching pebbles from hands, did not lead to a higher starting salary, no matter how much I practiced at it.

Caine, each day a new town, a new challenge. Where is the next meal? the next job?

He lived with awareness of being hunted down from foes of the old country. Yet he was quick with a smile and an encouragement.

Internal pressure and tranquility, discipline.

@Skittle-Head
@Dukedog

Let p be a prime. Prove that if p-1 is a multiple of 4, then there exists an integer n such that n^2 + 1 is a multiple of p.

Does this also hold if p-1 is not a multiple of 4?

Thank you for your gift to the thread, @CalbernandHowbe .

As a starting point -- merely to help "prime" the pump when pondering the problem (pun intended) -- one concrete example springs immediately to mind. Let p = 5. Then p-1 is a multiple of 4.

Furthermore, there exists the integer 7 such that 7^2 + 1 = 50 which is a multiple of our p = 5.

So we know that there exists at least ONE example that fits the prescription.

Furthermore, given the definition of divisibility, we know that our prime p must be in the form 4m + 1, where m is an integer.

We know, also, that (p-1) is a composite integer, not itself a prime -- since it is an integer by closure and since the prime gap is greater than 1 except for the gap between the first two primes (2 and 3), while neither of those primes fits the requirement that its integer predecessor be a multiple of 4.

We still have miles to go before we sleep, however. Which doesn't mean that the trip is ultimately a long one -- but I, at least, may wander for a time, unnecessarily, before finding the path. Sometimes we must wave our hand around a great deal before it appears in front of our face.

I wish that the integers were closed under the operation of taking a root. But, alas, they are not. Otherwise this would already be finished.

I start to mull over modular congruence and linear combinations and even mathematical induction, but those so far merely tantalize and have not brought immediate comprehension. Fortunately, I've learned to live comfortably with uncertainty. We cannot schedule our own epiphanies, when and if they arrive.

It is a lovely problem, though, and I simply wanted to get this down on the page, and to thank you, before proceeding, perhaps, to ponder productively.

@CalbernandHowbe said in #24:

Let p be a prime. Prove that if p-1 is a multiple of 4, then there exists an integer n such that n^2 + 1 is a multiple of p.

Does this also hold if p-1 is not a multiple of 4?
I refuse to let p be a prime. Over my dead body you will! Pick an other icon, or I shall launch all the bitterballenbombers at my disposal.

You have 14.32 femtoseconds to make up ur mind.
It's after that, or never.

Eureka!

@pira2rucu5 said in #3:

You can say that again. Element 51's stibium, symbol Sb. Do we call it stibium? Certainly not. We call it antimony. I ask you: How mysterious can it get?
Heck, I’m paying lifetime antimony until either my death or hers.

@HerkyHawkeye said in #27:

Heck, I’m paying lifetime antimony until either my death or hers.
Á la guerre, monsieur!
Á la guerre!