#factor theorem

[math-journal2]

<— Unit 20 — Unit 21 Poly — Unit 22 —>

Unit 21: Poly Functions Zeros

Part 1 —>

Remainder Theorem

#1

Factor Theorem

#1

Rational Zero Theorem

#1

Page 52

[eli-is-an-idiot]

how have i done so much work this week and yet i feel like i've learned nothing

[exopelagic]

talking to him more very much achieved. we just talked for like 4 hours in the kitchen holy shit I need to sleep

[math1089]

Mathematical Beauties of the Number 142587

There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.Leonhard Euler Welcome to the blog Math1089 – Mathematics for All. I’m glad you came by. I wanted to let you know I appreciate your spending time here on the blog very much. I do appreciate your…

[damneddamsy]

FALLING. RATING Explicit (18+ only) PAIRING Joel Miller x BIPOC OFC (Leela) FORMAT & SETTING Joel's POV & Post-TLOU Jackson AU WORD COUNT PER CHAPTER approx. 12,000+ STATUS Complete

SUMMARY It is said that every saint has a past, and every sinner has a future. Now, Joel Miller wasn’t looking to be a saint. Trust was a liability. Love, a memory too painful to keep. But if a sinner like him still had some future, and if that future starts with one night—a baby’s relentless cries cracking through his walls and breaking him open—then maybe, just maybe, he hadn’t lost everything yet. Against all instincts, he steps into that big, white house across his street. Nothing drives Joel to linger, but he does. For the baby at first—nascent Maya, with her bright eyes and fistfuls of Joel’s collar. Then, the strange new mother. What begins as an uneasy coexistence grows into something deeper, which neither of them dares name. Haunted by a narrative she never chose, brilliant but reclusive, Leela’s mind runs into the theoretical—proofs, patterns, chasing solutions to an unsolvable equation—while Joel’s hands are scarred by the practical: protecting, killing, enduring. When that peace becomes fleeting, when a fragile hope in the shape of a mathematical discovery begins to bloom, and the world, as always, threatens to take it away, Joel confronts what it means to fall—not just into the impossible, but into love, into hope, into the fragile rhythms of Leela and Maya’s life, and their quiet home that becomes a rare thing in this decaying tomorrow: a reason to stay. This is a story of healing, found family, and the abnormal, slow math of love—how we factor grief, multiply hope, balance the unknowns, it never adds up but somehow makes perfect sense.

INDEX (might be subject to change as the story progresses.)

part i -> EVENT HORIZON

part ii -> MICROFRACTURE

part iii -> FALSE EQUILIBRIUM

part iv -> MINIMUM VIABLE HOPE

part v -> RECONSTRUCTION ALGORITHM

part vi -> LIMIT APPROACHES GRACE

part vii -> FREEFALL FUNCTION

part viii -> SOFT INFINITY

part ix -> STITCH THEORY

interlude

part x -> DECOHERENCE

part xi -> ZERO CROSSING

part xii -> THEOREM OF BECOMING

part xiii -> HEURISTIC BLOOM

part xiv -> THE FINAL INTEGRATION

epilogue

acknowledgements

FALLING MOODBOARD (a huge bear hug, thank you and shoutout to the incredible @jolapeno !!)

FALLING MOODBOARD (2) (so many kisses and so much love to the talented, sweet @mrsmando !!)

CHARACTER STUDY A deep dive into Joel, Maya, and Leela, answering an ask from one of my sweetheart friends @jodiswiftle who followed along!

AUTHOR'S NOTE Have loads of fun with this masterlist! took me a while to think up a different way to potray these chapters, I'm so glad it came through so great!

TAGS your (ultimate) fix-it fic, The Dad™️ Joel, softest Joel you've ever seen, he is also an old yearner cuntstruck hardass, Joel being down bad for a teeny baby girl, OFC is arabic, OFC being an academic nerd and STEM girlie, the cutest baby (Maya) ever, baby is an actual character, Miller family dynamics, Tommy-Joel-Ellie hooliganisms, life in Jackson town, Ellie being the generally awesome older sister, neighbours-to-lovers trope, found family, slowburn, a lot of math references, lotsa door metaphors, epistolary interlude.

CONTENT WARNINGS eventual smut (the whole kaboodle), big griefs, depression, unbearable angst, violence, gore, blood, alcoholism, substance abuse, post-natal depression, the pains of motherhood, mentions of rape and suicide, childbirth.

[positively-knotted]

Time for me to abuse my access to peoples opinions again

[canisvesperus]

Calculus is significantly easier if you’re SUPER sharp on your trig and your algebra skills. If you know your identities and your theorems and you’re good at factoring and playing with numbers in general— you have a lot less to think hard about and trip up on while you’re working on the actual calculus itself, which can be surprisingly straightforward as long as you’ve trained yourself to see the patterns, stay organized, and think a few moves ahead.

[scarletwritesshit]

🦕 Anaxa x Aglaea 🦕 Wire Monkey Theorem

Aglaea always thought that Anaxa’s obsession with the dromases was rather frivolous. In fact, it was almost comical to her. Dromas figures, dromas plushies, dromas pajamas…dromas nicknames, even.

Despite a scoff here and there from Aglaea, it never bothered her, truthfully. A love for such creatures, though common and utilitarian, inflicted no harm. There were far more pressing matters for her to pick fights with him over, such as, literally everything. Everything, they conflicted over, until the very end.

A short-lasting resolution, only for her beloved Anaxagoras to be the next to fall victim to the hands of fate. Though she was the one to set the stones in place on Anaxa’s path to the final destination, it was not done so without hesitation. So many words were left unsaid, and though they had managed to find a common ground and stand upon it together, lingering grudges cannot be settled as fast as one may wish.

Anaxa had not even a gravesite for Aglaea to visit in memoriam. Only one of many constellations in the Amphoreus sky was reminiscent of his legacy, and her only present option was to speak to these very stars and pray that he could hear.

“Anaxa, my dear Anaxa, your silence pains me so. Your sharp tongue has grown dull; humor me with one final witty remark, won’t you please?”

No response.

It was to be expected, as stars don’t have a voice of their own. Nevertheless, Aglaea felt uneasy with the lack of retribution. Never would her threads tremor at the jab of an insult under his breath, or catch a glimpse of him walking amongst the streets of Okhema.

The illuminated constellation would still be there to gaze upon her from the sky.

Aglaea wouldn’t be able to see his guiding light.

Remnants of his studies still lay scattered about the Grove of Epiphany.

Aglaea could not read the calligraphy for how it was written by him.

All she had left to cling onto were the memories of his voice slowly fading from what remained of her mind and the sensation of when their hands would accidentally brush amidst their scuffles. The one man who had managed to make her feel almost human again would be there to prod her no longer. If she claimed to have felt empty before, then at the present moment, she was nothing more than a shell of a former human being.

Anaxa was the only person Aglaea felt as if she could cling onto, regardless of her willingness to admit such. What little that was left of him was just enough to patch up the void within her heart. She forced herself to turn away from the stars that she could neither see nor embrace in her arms.

What living love of Anaxa’s that did remain, however, were the dromases. The creatures very much served their purpose of transportation, and thus Aglaea thought positively of them alone, but she had never bothered to interact with them much for herself.

On the contrary, Anaxa was obsessed with them. She had seen his obsession as childish, and a complete waste of time, though now the dromases were perhaps her only way of connecting with him at least one final time.

The dromases were not an uncommon occurrence within the city walls, a partial factor as to why she could not understand his fascination with them. Or perhaps, their commonness was indeed a factor as to why he was so obsessed with them.

Aglaea wished that she had asked him why. Just once.

Too late.

She had nothing left.

Just what he had left.

⋆｡°✩

She ran the palm of her hand down the side of the dromas's thick leg. Its scales were smooth and radiated a warmth existent only due to the ever-present sun.

In a way, it reminded her of Anaxa.

Everything was reminding her of Anaxa.

The massive creature kept its leg still, careful as to not accidentally crush her. Aglaea was prepared to duck out of the way in the event of a sudden movement, but the beast was surprisingly mindful of her smaller size. As she was gently stroking its leg, she felt the creature sniff her gently, then groaned out of curiosity.

"You still smell remnants of your beloved Anaxagoras on me, don’t you?"

The dromas growled gently and quietly. She had never bothered to communicate with them so closely, but already she could feel the sense of a mutual understanding. It nudged her very gently with its nose, as if to say yes to her question.

"You’re such an adorable thing. I would almost hate to say these words to you myself, though it’s not like you can understand me."

The dromas shook its head as if to say, Of course I can understand you, my lady! Aglaea smiled, then rested her head against its leg.

"Oh, you poor beast. Anaxa has regretfully departed from this world. He couldn’t even say his final farewell to any of you.”

The dromas growled, lifted up its head, and shook it no in disbelief. Aglaea could feel through her golden threads that the dromas was quite upset. It clearly did not want to believe her, but it was hard to dispute the integrity of her words when her body held such strong traces of his scent. She hadn’t enough experience with the dromases to be able to personally comfort them, let alone the single one she was leaned against.

The ones in the pen cried out once the realizations had struck them. Those who were strolling by stalled their travels in equal disbelief. Aglaea was being deafened by their grief vibrating throughout her strings. And she had no way of knowing how to silence it, thanks to her negligence.

It really seemed as if the dromases were more than just a strange fascination to Anaxa. The creatures viewed him as one of their own, and the love he held for them was all the equal.

Aglaea felt alive, but only alive with guilt.

Though unable to communicate in the same tongue, the dromas that Aglaea had clung onto had offered its condolences.

"I should be offering you my apologizes," she said, "for I was ignorant of just how much he cared for your kind."

The dromas seemed to understand that there was no harm nor foul intended by her actions. It nudged her as if to say it accepted and understood her apology. The threads that bound their communication together remained motionless. Could they even serve as a reliable indicator of a creature’s thoughts? She hadn’t bothered to question such prior. Aglaea would have to hope that the treads that bound the city were as reliable as always, upholding the same values of truth between Chrysos Heir and dromas alike.

It seemed to have accepted her rather quickly. Perhaps knowing of Anaxa’s scent helped her gain its trust. Aglaea let go of the dromas, and it protested by nudging its leg towards her. It seemed to be wanting her to act as a stand in for Anaxa.

She couldn’t do it. She didn’t even have a fraction of the knowledge as he did.

She had nothing left.

Not even her own humanity left.

Aglaea wasn’t even sure how much longer she was going to be physically around herself.

"I must leave now," she said, gently nudging the dromas away with the palm of her hand, "my duty calls, and the prophecy waits for no one.”

Unable to look back, Aglaea left the dromas pen. The dromas stomped its foot and bellowed for her return, but ultimately gave up and turned away.

During a rare moment of downtime, Aglaea once more paid a visit to the dromases. The gentle roar from the creature signified that it was delighted for her return.

"I know nothing, yet you are still delighted to see me?"

The dromas groaned in agreement.

"You are most aware that I cannot substitute as him, correct?" she said, lifting her hand up.

The dromas nudged her hand gently with its nose. It had grown quite fond of her touch rather quickly. A most curious behavior as her hands hardly felt the same as his, her body hardly had the same chill as his, and her body would no longer carry remnants of that scent of his.

It must be terribly lonely, Aglaea thought.

The others were eyeing her up with longing as well, and she could sense a void within their hearts that felt all too familiar. Sliding her hand off of the dromas’s nose, she walked over to the other ones within the enclosure, who looked at her with immense curiosity. They took turns investigating her, and all it took was a single sniff for them to accept her.

Anaxa’s presence must have provided them with a man and beast friendship that only he could bring, and they must have been under the assumption that Aglaea was capable of doing the same.

They were quiet.

They were gentle.

It wouldn’t wear away at her to spend even just a little extra time with the dromases.

"...I can at least come and visit you, should time allot to it."

The dromases stared at her in anticipation.

"Though, do be aware that I cannot serve as a true replacement for our beloved Anaxagoras."

Despite the dromases not being satisfied with her answer, they still retained a mellow tone. No true protests, nothing. Aglaea was starting to understand why he was so greatly fond of those creatures.

Though there was a mutual respect between Chrysos Heir and creature, something still felt...missing from their bond. The scales of the dromases naturally felt cool and smooth like Anaxa’s shell of a body, but they didn’t quite feel like him. Remnants of Anaxa were still present on Aglaea, but even the dromases weren’t blind to the fact that she wasn’t him.

All in all, it was a temporary solution. A bandaid laid upon shattered pottery. Even when such a realization fell upon Aglaea, she still continued to selfishly dedicate her time to the dromases, as a stand in for the comfort of Anaxa’s presence. As time carried on, despite Aglaea and the dromases having each other, it only seemed as if they all grew lonelier and emptier. She didn’t allow herself consciously to acknowledge that.

"It’s just as how Anaxa would have carried on," Aglaea claimed.

The strings in Okhema vibrated.

[lipshits-continuous]

Explanation Posts

I explain my url (continuity in ℝ) 05/06/2022

Definition of a Ring 07/10/2022

Definition of a Metric Space 16/10/2022

Open and Closes sets in a Metric Space 16/10/2022

Convergence and Continuity in Metric Spaces 22/10/2022

Bounded and Compact sets in a Metric Space 23/10/2022

I show 0.9999...=1 30/11/2022

Linear Maps and Isomorphisms 18/01/2023

Ring homomorphisms, Ideals, Quotients, and the First Isomorphism Theorem 12/03/2023

Mutamorphic Functions (technically not an explanation post but still belongs here) 29/06/2023

Second Countability of ℝⁿ 04/07/2023

The Weierstrass Function 23/08/2023

Unbroken Workbrooks (A post about continuity written in Anglish) 19/09/2023

Countable Sets have 0 Lebesgue Outer Measure 17/10/2023

Introduction to Homotopy 05/02/2024

The Fundamental Group 16/02/2024

Continuity in Terms of Open Sets 16/02/2024

The Set of Finitely Generated Free Groups as a Monoid 01/03/2024

Intro to Topology: Metric Spaces 12/03/2024

Intro to Topology: Topological Spaces and Continuity 19/03/2024

Intro to Topology: Closed Sets and Limit Points 26/03/2024

Intro to Topology: Hausdorffness 26/03/2024

Intro to Topology: Connectedness 27/03/2024

Intro to Topology: Path Connectedness 28/03/2024

Intro to Topology: Compactness 11/04/2024

Intro to Topology: Bases and Second Countability 12/04/2024

Intro to Topology: Product Spaces 23/04/2024

Intro to Topology: The Heine-Borel Theorem 24/04/2024

Intro to Topology: Quotient Spaces 25/04/2024

Intro to Topology: Important Examples 26/04/2024

Intro to Topology: Conclusions and Remaining Questions 26/04/2024

The Dirac Delta Distribution is not an Extended Real Function 03/05/2024

The Fundamental Group of a Topological Group is Abelian 10/06/2024

Proof of the Factor Theorem 30/06/2024

A Short Intro to Category Theory 23/10/2024

An Example of When Connectedness Implies Path Connectedness 17/11/2024

Relative de Rham Cohomology 13/06/2025

[michaelrotonal]

What is your favorite theorem? Also, do you have any controversial opinions about homology?

my favorite theorem is probably that every number in A174973 (OEIS sequence; every number whose factors increase by a factor of at most 2) is possible in SQUART (my favorite mode in The 2048 Power Compendium; you may add two numbers if the smaller number's square is a multiple of the larger number). it's my favorite because i proved it myself

and for the second question the answer is no

[prime-factorizer]

what is a prime factor

i like your funny numbers but i dont know what they mean :(

First of all, we'll only be talking about positive whole numbers, so no decimals, fractions, complex numbers, etc. This is also just going to be an introduction because this topic is more complicated/nuanced than you might expect, and I don't know/understand a lot of it

With that out of the way prime number is a number that you can't get to by multiplying other numbers: 24 isn't prime because you can write it as 6×4, but 23 is prime because the only way to get to 23 (only using multiplication) is 1×23. If you're interested, I encourage you to try finding a few prime numbers. It can be a fun exercise if you haven't tried it before. I'll put all of the ones less than 100 under the cut

A prime factor is just a prime number that divides another number. You can divide 24 by 3, and 3 is prime, so 3 is a prime factor of 24

Also, if you can get to a number by multiplying other numbers, then it's a composite number. Also also, 0 and 1 are weird so just don't worry about them <3

The thing that makes this blog work though is what's known as the fundamental theorem of arithmetic (ftoa):

Like I said, you can write 24 as 6×4, but you might notice that neither 6 nor 4 are prime. If you rewrite 6 as 2×3 and 4 as 2×2, then you can then write 24 as 2×3×2×2. Now all of the numbers are prime!

But you'll notice you can also write 24 as 8×3, then write 8 as 2×4 to get 2×4×3, then write 4 as 2×2 to get 2×2×2×3. Now that all of the numbers are prime, you'll notice that they're the same as when we started with 6×4 (three 2s and one 3)

What the ftoa says is, no matter how you break down a number like this, you'll always end with the same prime numbers at the end (in this case, three 2s and one 3)

The only thing I do differently from this process is use exponents to make it shorter. In this case, you can rewrite 2×2×2 as 2³. So, when I say 24 is 2³×3, I've "factorized" 24, and 2 and 3 are its prime factors. Of course, I don't usually do this by hand or in my head; I just give the number to a computer and it does it for me (it can be fun to do it yourself though, so sometimes I'll do it in my head for the smaller numbers)

Prime numbers and the ftoa play a pretty big role in math, and there's a ton of unsolved problems related to them, which is why you'll see people make a big deal out of them

Prime numbers under 100 and other notes under the cut:

Prime numbers under 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

0 and 1 aren't considered prime or composite. There's probably a more sophisticated answer to why, but I like to justify the decision by saying they ruin the ftoa:

0 is divisble by every number, so it's not prime but you also can't factorize it, so it isn't composite. 1 isn't prime because if it was then you could add as many ×1s as you wanted to a factorized number, ruining the uniqueness of its factorization, and, once again, it also can't be composite because you can't factorize it

Also, like I said, this is only an introduction and I don't understand most of the more complicated/nuanced definitions. If you wanna learn about those though, I recommend TheGrayCuber's videos:

[math-journal2]

Factor theorem

[rbrooksdesign]

"DMT_45," digital + acrylic, May 17, 2024, Reginald Brooks

DMT = Divisor (Factor) Matrix Table

Divisors of 224:

Row1 (6 cells): 1-2-4-8-16-32

Row7 (6 cells): 7-14-28-56-112-224

Why are 224-112-56-14 not each a Perfect Number (PN), yet 28 is?

If one sums up of their respective factors, only does 28 equal itself -- 1+2+4+7+14=28. (Here factors = divisors - number itself.)

PN28 is also part of the Mersenne Prime-Perfect Number pairings, of which there are only 51 currently known.

PN=2ᵖ⁻¹ (2ᵖ -1) = 2²(2³ -1) = 4(7) = xz = xMp, where p=Prime, Mp=Mersenne Prime. The first part is the Euclid-Euler Theorem.

more...

[o2studies]

༻`` 5 Apr 24 — Friday 1/60

🧡 academic >> I finished questions for math. It felt great to get into them and they did get very repetitive by the end but I at least now have covered the whole chapter and know I can get full marks at almost every question (topic: remainder and factor theorem)

🧡 physical health >> Did the slightest bit, lifted a couple weights a few times but I wasn't going to do anything so that's still an improvement

🧡 self care >> I'm going to set my alarm now so that I won't have to look at my phone after I dry my hair before bed. I also set a screen time goal of 2hrs 40 because for these past 2 weeks my screen time has gotten up to 5-7 hours a good few times... I can feel how tired my eyeballs currently are... Also drew on myself a lot today and wore mismatched socks which improved my mood a whole lot ^^

Did not do much at all but I did have a much better mood today and I ate well enough. If I remembered about this challenge earlier than 9pm I would've pushed myself to do more.

[softpine]

the amount of knowledge in guns you need to have to weave that into the story is fascinating to me. I don’t even know what racking a gun is (I do however know a whole bunch about obscure physics theorems) anyway I just think it’s neat how far we go into research to write things that make sense

i'm still not sure if i got everything correct, but no one has told me otherwise fjkjsd you've definitely heard the sound of a shotgun racking if you've watched any movie with guns in it, it's that "ch-chuck" kinda sound. what i didn't know is that there's apparently a huge debate in the home defense gun community over whether you should keep your gun loaded and racked or just loaded (don't bother mentioning safeties, they'll call you slurs...). the true benefit of not racking the gun would be if a little kid gets a hold of it, they're not going to know how to rack it therefore even if they pulled the trigger they hopefully wouldn't be able to shoot anything (in MY opinion this is idiotic; the chance that the gun would go off anyway is never zero, especially if they've ever seen their parents shooting and decide to copy their movements, as kids love to do). then some people think you should wait to rack it until you know there's an intruder in your house because the mere sound of it will make them go "oh shit" and just leave. people on the other side of the argument think this is pointless and only serves to reveal your position (but uhhh fyi, in some states you're supposed to give them a verbal warning and/or try to escape before using deadly force anyway). and they say you shouldn't be owning a gun if you're not prepared to use it as anything other than a sound effect. plus i saw some people say that you need that 1 extra round in the chamber or you might run out, which.... most pump shotguns hold 3 - 5 rounds so.. it just seems excessive to me, i can't imagine shooting up to 5 times and missing all of them?? in that case dude i think you're just fucked and 1 extra try is not going to help you lmao

the time factor is the most compelling argument to me, obviously if you're asleep and you hear someone break in and enter your bedroom, your gun is useless unless it's ready to shoot right this second. but the chances of this happening are just so slim and you know what would be even more effective?? a home alarm system so no one can possibly make it to your bedroom without you knowing about it. i saw a lot of people say they keep a loaded shotgun under their bed for this scenario. i'm a paranoid fuck and i've always lived in areas with high crime rates so i get it, but that's why i keep a maglite next to me while i sleep so in the incredibly unlikely event that someone breaks in without me hearing it, i can you know. whack them. do i ever think i'll need to do that? no but it makes me feel safer than having a loaded and chambered gun with no safety on lmao that would make me feel incredibly UNsafe... also i don't want to make this too dark but while i was in high school i knew FIVE kids who died because of their parents guns, one way or another. that is just unacceptable to me. we also had a gun go off at school and even though no one was hurt, that was one of the most traumatic experiences of my life because obviously in the moment we all heard a gunshot and thought it was an intentional school shooting. so anyway, me and guns do not get along but it's fascinating to research

but yeah as you can see i fell down the rabbit hole!! the things you do for writing jfksjds i did not need to go that far with it, but it was interesting!

[spacetimewithstuartgary]

New method of measuring gravity with 3D velocities of wide binary stars is developed and confirms modified gravity at low acceleration

Wide binary stars with separation greater than about 2000 astronomical units are interesting natural laboratories that allow a direct probe of gravity at low acceleration weaker than about 1 nanometer per second squared. Astrophysicist Kyu-Hyun Chae at Sejong University (Seoul, South Korea) has developed a new method of measuring gravity with all three components of the velocities (3D velocities) of stars (Figure 1), as a major improvement over existing statistical methods relying on sky-projected 2D velocities. The new method based on the Bayes theorem derives directly the probability distribution of a gravity parameter (a parameter that measures the extent to which the data departs from standard gravitational dynamics) through the Markov Chain Monte Carlo simulation of the relative 3D velocity between the stars in a binary.

For the significance of the new method, Chae says, “The existing methods to infer gravity have the limitation that only the sky-projected velocities are used. Moreover, they have some limitations in accounting for the uncertainties of various factors including stellar masses to derive the probability distribution of a gravity parameter. The new method overcomes all these limitations. It is a sort of revolutionary and ultimate method for wide binaries whose motions can only be ‘snapshot-observed’ (that is, observed only at a specific phase of the orbital motion: because of the very long orbital periods of these binaries, a direct consequence of the low accelerations involved, one can only measure the positions and velocities of the stars at one moment, which is far less informative than having, ideally, data on a full orbit or at least a segment of it).” Chae adds, “However, the new method requires accurate and precise values of the third velocity component, that is, the line-of-sight (radial) velocity. In other words, only wide binaries with precisely measured radial velocities can be used.”

On the significance of the methodology, Xavier Hernandez, who initiated wide binary gravity tests in 2012, says, “The latest paper by Dr. K.-H. Chae on wide binaries presents a fully rigorous Bayesian approach which will surely become the standard in the field. Further, this latest paper presents also a proof of concept in going from 2-dimensional projected velocities to full 3D relative velocities between the two components of a wide binary. The level of accuracy reached from making full use of all available information is impressive.”

For the first application of the new method, Chae used about 300 wide binaries with relatively precise radial velocities selected from the European Space Agency’s Gaia data release 3. Although the first results are limited by the fact that Gaia’s reported radial velocities are not as precise as the sky-projected velocities, the derived probability distributions of gravity agree well with the recent results published by Chae and independently by Hernandez’s group as well. For wide binaries whose stars orbit each other with an internal acceleration greater than about 10 nanometers per second squared, the inferred gravity is precisely Newtonian, but for an internal acceleration lower than about 1 nanometer per second squared (or separation greater than about 2000 au), the inferred gravity is about 40 to 50 percent stronger than Newton. The significance of the deviation is 4.2sigma meaning that standard gravity is outside the 99.997 percent probability range. What is striking is that the deviation agrees with the generic prediction of modified gravity theories under the theoretical framework called modified Newtonian dynamics (MOND, sometimes referred to as Milgromian dynamics), introduced about 40 years ago by Mordehai (Moti) Milgrom.

On the first results based on the new method, Chae says, “It is encouraging that a direct inference of the probability distribution of gravity can be obtained for wide binaries that are bound by extremely weak internal gravity. This methodology may play a decisive role in the coming years in measuring gravity at low acceleration. It is nice that the first results agree well with the results for the past 2 years obtained by Hernandez’s group and myself with the existing methods.”  

Pavel Kroupa, professor at the University of Bonn in Germany, says, “This is an impressive study of gravitation using very wide binaries as probes taken to a new level of accuracy and clarity by Prof. Dr. Kyu-Hyun Chae. This work greatly advances this topic, and the data, which will be improving over time, are already showing an increasingly significant deviation from Newtonian gravitation with an impressive consistency with the expectations from Milgromian gravitation. This has a major fundamentally important impact on theoretical physics and cosmology.”

Milgrom expresses his thoughts on the general significance of the wide binary results.  “This new result by Prof. Chae strengthens in important ways earlier findings by him and others. They demonstrate a departure from the predictions of Newtonian dynamics in low-acceleration binary stars in our Galaxy. Such a departure from standard dynamics would be existing in itself. But it is even more exciting because it enters and appears in the same way as the departure from Newtonian dynamics appears in galaxies. It appears in the analysis only at or below a certain acceleration scale that is found to agree with the fundamental acceleration of MOND, and the magnitude of the anomaly they find is also consistent with the generic predictions of existing MOND theories. In galaxies, the observed (and MOND-predicted) anomaly is much larger, and is established very robustly, but much of the community support the view that it is due to the presence of dark matter; so, to them the galactic anomalies do not bespeak a conflict with standard dynamics. But, an anomaly as found by Prof. Chae, while more modest, cannot be accounted for by dark matter, and thus would indeed necessarily spell a breakdown of standard dynamics.”

Chae and his collaborators including Dongwook Lim and Young-Wook Lee at Yonsei University (Seoul, South Korea) and  Byeong-Cheol Lee at Korea Astronomy and Space Science Institute (Daejeon, South Korea) are now obtaining precise radial velocities from their new measurements using observation facilities such as GEMINI North Observatory (with the instrument MAROON-X) and Las Cumbres Observatory, and from archival data outside Gaia as well. Hernandez and his collaborators are carrying out the speckle photometry of target wide binaries to identify any systems with a hidden third star. Hernandez comments on this point, “This methodology requires using pure binaries that are free of any hidden companion stars. This highlights the relevance of upcoming results from dedicated ground-based follow-up studies which will unambiguously rule out dubious systems containing hidden third components and hence permit to reach the full potential of the new method.”  When all these observation results are combined, decisive results on the low-acceleration anomaly are expected.

On the near future prospect Chae says, “With new data on radial velocities, most of which have already been obtained, and results from speckle photometric observations, the Bayesian inference is expected to measure gravity sufficiently precisely, not only to distinguish between Newton and MOND well above 5sigma, but also to narrow theoretical possibilities of gravitational dynamics. I expect exciting opportunities for theoretical physics with new results in the coming years.”

IMAGE:  3D velocities versus sky-projected 2D velocities of a wide binary system. The new method uses the 3D velocities while all existing methods use the 2D velocities. Credit Kyu-Hyun Chae