@JohnsonFido @capobianco_slv @GWOMaths Got it.

a² - d⋅b² = ±4

we have, equivalently
(½a)² - d⋅(½b)² = ±1

(½a - √d⋅½b)³
= a⋅½(db²±1) - b⋅½(a²∓1)⋅√d

where both of
are always even for d, a, and b odd.

But for d ≅ 1 (mod 4)
then a and b are always the same parity.
@JohnsonFido @capobianco_slv @GWOMaths Thus
x = a⋅½(db²±1)
y = b⋅½(a²∓1)

are positive integer solutions to
x² - d⋅y² = ±1

a² - d⋅b² = ±4

for positive integers a, b.

I believe this is only relevant for d ≅ 1 (mod 4) even though that condition doesn't appear in the calculation.
@JohnsonFido @capobianco_slv @GWOMaths Derivation:

= (½a - ½b⋅√d)³
= (½a - ½b⋅√d) ⋅ ¼⋅(a² + d⋅b² - 2ab⋅√d)
= ⅛⋅(a³+da⋅b²+2⋅da⋅b²) - ⅛⋅(a²⋅b+d⋅b³+2a²⋅b)⋅√d
= ⅛⋅(a³+3da⋅b²)- ⅛⋅(d⋅b³+3⋅a²⋅b)⋅√d
= ⅛⋅(4da⋅b² ± 4) - b⋅⅛⋅(4⋅a² ∓ 4)⋅√d
= a⋅½(db²±1) - b⋅½(a²∓1)⋅√d

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More from @pgeerkens

17 Jan
@johnrobb We have abandoned competence,
in the futile search for universal expertise.

"What?" I hear you say.
Clearly definitions and explanations are in order.

@johnrobb Expertise, certainly as regards this discussion,
but perhaps also universally, should be regarded
as the knowledge of a very great deal - about very
very little.

@johnrobb To become an expert in a field requires
years - nay, decades - of diligent study of that
solitary field. In diving deeper, the pool in turn
becomes ever smaller: a diving pool rather than a
swimming pool. Nothing is free.

Read 9 tweets
15 Jan
@GWOMaths One might say that physicists study the symmetry of nature, while mathematicians study the nature of symmetry.

@GWOMaths Observing symmetry in nature, such as noting the similarity between the symmetries of a snowflake and a hexagon, is readily comprehensible. What does it mean then to study "the nature of symmetry"?

@GWOMaths Mathematicians define a "group", G, as a set of elements {a,b,c, ...} with a binary operation ⊡ and a distinguished element e (the identity of G) satisfying these specific properties:

Read 14 tweets
2 Dec 20
@pnjaban None of the above:
Field-trained former military paramedic.

I don't want anyone not trained specifically in emergency medicine anywhere near; and none of those civilian trained E.R. either, cause they're trained in CYA first, and I want someone who's serious about saving my life
@pnjaban When my Dad suffered acute hemechromatosis at ~34,
for which the ONLY treatment is *bleeding*,
and the only options "a whole lot" and "a whole lot more",
the attending physician was too embarrassed to prescribe the correct treatment.
@pnjaban The nurse (Irish of course, because they understand hemechromatosis) told me Mom:

Mrs. Geerkens:
Your husband will die tonight
unless we remove him from this hospital immediately.
I'll help.

The nurse lost her job;
but my Dad lived another 45 years.
Read 4 tweets
2 Dec 20
From Albert H. Beiler's
Recreations in the Theory of Numbers:

Calculate in your head:
47² = ?
96² = ?
113² = ?
179² = ?

47² = (47 + 3) ⋅ (47 - 3) + 3² = 50 ⋅ 44 + 9 = 2209

96² = (96+4) ⋅ (96-4) + 4² = 100 ⋅ 92 + 16 = 9216

113² = (113+3) ⋅ (113-3) + 3² = 116⋅110 + 9
= 12760 + 9 = 12,769

179² = (179+21) ⋅ (179-21) + 21²
= 200 ⋅ 158 + (20² + 20 + 21)
= 31600 + 441 = 32041
The trick is from noting that from difference of squares
a² - b² = (a + b) ⋅ (a - b)
one can obtain by rearrangement
a² = (a + b) ⋅ (a - b) + b².
Read 4 tweets
22 Nov 20
@GWOMaths Hint follows
@GWOMaths Rochambeau is just a fancy name for Rock-Paper-Scissors.
@GWOMaths If you're having difficulty visualizing this,
consider the situation after Game 1 as the baseline.
Read 12 tweets
6 Nov 20
@LarrySchweikart My take:
SCOTUS must not be seen to play favourites - yet has an obligation to ensure that corruption is voided.

So wherever widespread and continuous failure to observe due process (as per State law) is seen, void all ballots from that county / counting centre.

@LarrySchweikart If the number of voided counties exceeds more than 1 or 2, then void the entire election for the State.

In the case of House and Senate representatives, require special elections.

@LarrySchweikart For Presidential Electors, legislation now in effect gives the State legislature responsibility and authority to select the State's slate of Electors - as per the original Constitutional design.

Read 9 tweets

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