(from GPT)

Milestone results (with quick why-they-matter)

・Dynamic programming origin story — Bellman’s DP framework that underlies the classic pseudo-polynomial knapsack DP. (Google Books)
・NP-completeness — Karp includes (0-1) Knapsack among his original 21 NP-complete problems (1972). This set the hardness backdrop for everything after. (cgi.di.uoa.gr)
・First FPTAS for 0-1 Knapsack — Ibarra & Kim (1975) kicked off the approximation era; later refinements followed. (SpringerLink)
・Algorithm-engineering bible (exact & approximate) — Martello & Toth’s monograph (1990). Still the go-to for classic branch-and-bound, DP variants, and benchmarks. (doc.lagout.org)
・Modern comprehensive reference — Kellerer, Pferschy & Pisinger (2004/2013) surveys the whole zoo of knapsack variants and techniques. (SpringerLink)
・Core-based exact methods — Pisinger (1997) “minimal/expanding core” made exact 0-1 knapsack solvers blisteringly fast in practice. (INFORMS Pubs Online)

Subset sum (the “pure math” spine) and lattice breakthroughs

・Meet-in-the-middle refined — Schroeppel & Shamir (1981): time (O(2^{n/2})) with only (O(2^{n/4})) space; a landmark time–space trade-off. (SIAM Ebooks)
・LLL meets subset sum — Lagarias & Odlyzko (1985): polynomial-time success on “low-density” instances via lattice reduction; later sharpened by Coster et al. (1992). This is where number theory and geometry of numbers bite. (ResearchGate)
・Near-linear pseudo-poly time — Bringmann (2016/2017): randomized (\tilde O(n+t)) for Subset Sum (target (t)), essentially tight under standard conjectures. Koiliaris–Xu (2017→2019) gave the fastest deterministic pseudo-poly. (arXiv)
・Recent worst-case advance — Nederlof–et al. (STOC’21): improves the classic Schroeppel–Shamir bound with a slick randomized algorithm. (ACM Digital Library)

Cryptographic detour (revealing the structure)

・Merkle–Hellman (1978) & Shamir’s break (1984) — Early public-key based on subset sum; Shamir’s polynomial-time attack (and later refinements) illuminated why low-density structure is fragile. (Wikipedia)

People to know (anchor researchers)

・Complexity & foundations — Richard Karp (NP-complete classification). (cgi.di.uoa.gr)
・Approximation pioneers — Oscar H. Ibarra, Chul E. Kim (FPTAS); Eugene Lawler (early approximation ideas). (SpringerLink)
・Exact algorithms / engineering — Silvano Martello, Paolo Toth; David Pisinger; Ulrich Pferschy; Hans Kellerer. (doc.lagout.org)
・Subset sum & lattices — Richard Schroeppel, Adi Shamir, Jeffrey C. Lagarias, Andrew M. Odlyzko, Marc Coster. (SIAM Ebooks)
・Modern pseudo-poly speedups — Karl Bringmann; Konstantinos Koiliaris & Chao Xu. (arXiv)

Handy “starter” reading list (one-line reasons)

・Karp (1972) — Why knapsack is hard in the worst case. (cgi.di.uoa.gr)
・Ibarra–Kim (1975) — First fully polynomial scheme; archetype of knapsack FPTAS. (SpringerLink)
・Martello–Toth (1990) — Classic cookbook of exact/approx methods that still holds up. (doc.lagout.org)
・Kellerer–Pferschy–Pisinger (2004/2013) — Encyclopedic modern treatment and variants. (SpringerLink)
・Schroeppel–Shamir (1981) — Time–space landmark for subset sum. (SIAM Ebooks)
・Lagarias–Odlyzko (1985) & Coster et al. (1992) — Lattices + low density = structural wins. (ResearchGate)
・Bringmann (2016/2017); Koiliaris–Xu (2017→2019) — Fastest pseudo-poly algorithms we have. (arXiv)

Summary of “PRIMES is in P” and Its Context

1. Background and Motivation

・Primality testing—determining whether a number is prime or composite—is a central problem in mathematics and cryptography.
・Naive trial division up to $\sqrt{n}$ requires $\Omega(\sqrt{n})$ steps, far too slow for large numbers.
・An efficient algorithm must run in polynomial time with respect to the input size, $\log n$.
・Since the 1970s, the problem was known to lie in $\mathbf{NP} \cap \mathbf{co\text{-}NP}$, but a fully deterministic polynomial-time algorithm was elusive.

2. Historical Developments

・Miller (1975): Provided a deterministic polynomial-time test assuming the Extended Riemann Hypothesis (ERH).
・Rabin (1980): Made the test unconditional but randomized, creating a probabilistic primality test.
・Adleman–Pomerance–Rumely (1983): Produced a deterministic algorithm running in $(\log n)^{O(\log \log \log n)}$ time.
・The long-term goal was an unconditional deterministic polynomial-time algorithm—finally achieved by AKS.

3. Core Insight of the AKS Algorithm

・Built upon a generalization of Fermat’s Little Theorem: for prime $p$, $a^{p-1} \equiv 1 \pmod p$.
・AKS uses the congruence:
$$(X + a)^n \equiv X^n + a \pmod n$$
・Directly checking this identity is too costly ($\Omega(n)$), so AKS checks it modulo $X^r - 1$ for a small, carefully chosen integer $r$.
・The test verifies the congruence for multiple small $a$ values.

4. Steps of the Algorithm

1. Check for perfect powers: Determine if $n = a^b$ for $b > 1$.

2. Find minimal $r$: Choose the smallest $r$ such that the order of $n$ modulo $r$, $o_r(n)$, exceeds $\log^2 n$.
   ・Such an $r$ always exists with $r \leq \max{3, \lfloor \log^5 n \rfloor}$.

3. Verify polynomial congruences:
   ・Check $(X+a)^n \equiv X^n + a \pmod{X^r - 1, n}$ for $a = 1$ to $\ell = \lfloor \sqrt{\phi(r) \log n} \rfloor$.

5. Mathematical Foundations

・The concept of introspective polynomials:
$$[f(X)]^m \equiv f(X^m) \pmod{X^r - 1, p}$$
・These are closed under multiplication, ensuring structural stability in the proof.
・Two algebraic groups underpin the correctness proof:
・$\mathbf{G}$ (number residues) with size $t > \log^2 n$
・$\mathbf{\mathcal{G}}$ (polynomial residues), for which Hendrik Lenstra Jr. provided a lower bound.
・The proof shows that if the algorithm outputs PRIME, then $n$ must be a power of a prime.

6. Complexity and Improvements

・Original asymptotic complexity: $O\sim(\log^{21/2} n)$, dominated by the polynomial checks.
・Improved to $O\sim(\log^{15/2} n)$ using sieve theory (primes with large $P(q-1)$).
・Under the Sophie-Germain Prime Density Conjecture, the heuristic complexity drops to $O\sim(\log^6 n)$.
・A refined Lenstra–Pomerance version achieves a provable $O\sim(\log^6 n)$ bound.

Scientific Management = Notion + GPT + ISO?

I believe scientific management can be the key to solving problems unique to solo founders.
My working hypothesis is that the combination of Notion, ChatGPT, ISO, and reverse management can address most of these challenges.

Three years ago, our team set a clear direction: to specialize in scientific and technological research that creates a big impact with a small group of people. I’ve continued this pursuit as a solo founder myself.
The hardest part is analyzing myself objectively.

Practical Method: Using Notion and “Letters”

Here’s a method I recommend:
Use Notion to write a daily “letter” addressed to each functional role in your company (e.g. marketing, engineering, finance).
Create a thread for each role weekly, and accumulate your reflections there.
Think of it like a Kanban system—one thread equals one task.
At the end of each week, throw the whole thread into ChatGPT and ask for analysis.

Logical Analysis and Feedback with ChatGPT

Ask GPT to “analyze the strengths and weaknesses of my management style logically and objectively, based on this content.”
Then add, “From the perspective of ISO (International Organization for Standardization), how would this be evaluated?”

Here, ISO doesn’t refer to certification but to the international standards for quality management.
The strength of ISO lies in its focus on quantitative representation—it lets you express processes and outcomes numerically.

Quantitative Goal Setting and Reverse Management

Applying ISO principles allows you to evaluate your behavior and results quantitatively every 12 weeks (one quarter).
You can even ask ChatGPT to “set an ideal vision and action plan for 12 weeks ahead based on ISO standards.”
This enables logical goal design rather than gut feeling.

By repeatedly analyzing and feeding back results, you can observe your own growth over time.
The focus shifts from mere outcomes to continuous improvement of the management process itself.

Making Management Quality Visible

This approach enables “ISO-based fixed-point observation” — a structured way to see how much you’ve grown in 12 weeks.
It’s essential not to focus solely on achieving business goals, but also to evaluate the quality of your management process.
Most people emphasize “motivation,” “slogans,” or “numerical goals,” yet overlook whether the process itself is improving.

The Value of Practice and Sharing

I’ve been experimenting with this method continuously and plan to share both its successes and shortcomings.
Now that ChatGPT costs have dropped, there’s no better time to apply it.

Scientific management means leading your business based not on intuition, but on quantitative and logical criteria.
By structuring in Notion, analyzing with GPT, and anchoring in ISO, this triadic approach enables both self-objectivity and sustained growth.

(科学的経営 = Notion + GPT + ISO?)

Notionで構造化し、GPTで分析し、ISOで軸を定める――この三位一体のアプローチによって、自己客観化と成長を両立できます。

(日本語下記)

The Pleasure of Creative Problem-Solving

This story connects to something like Nobita’s endless effort to find a way out of his own vague troubles. Regardless of whether one is from the humanities or sciences, I think this kind of continuous reflection is what people who invent or try new things truly love.

For example, when faced with problems such as persistent economic inequality or incurable diseases, it’s easy to think of ways to give up. You could probably come up with a thousand ways to justify doing nothing. That’s not a creative act—it’s more like an instinctive reaction. I don’t deny that those thoughts arise naturally, but solving truly difficult problems requires ideas so vast that even a million minds might not be enough. The theoretical space of possible solutions is practically infinite, and that’s what makes such challenges both maddening and exhilarating.

The Solitary Joy of Experimentation

If you think about it that way, conducting experiments on your own is extremely difficult. There are infinite theoretical solutions, but only limited time to explore them. That’s why the act of experimenting itself becomes a search for light within a narrow window of possibility. People like to say, “Follow the data,” or “Think outside the box,” as if it’s a slogan—but in practice, it’s never that simple. Tomita, twisting the knobs of his early synthesizers and hunting for sounds in darkness, was engaging in precisely that kind of solitary labor. In the end, the process of fumbling through uncertainty—of wrestling with frustration until a faint light appears—is what makes it so deeply fulfilling. And when it’s over, one realizes that the time spent struggling was, in fact, the most enjoyable part.

Rational Companions and a Resistant World

The people who empathize with such new attempts are usually rational thinkers. Emotional audiences often say things like, “Don’t bother with that,” or “It’ll never work.” Others insist that “success comes from imitation.” But creativity doesn’t mean going against others for the sake of rebellion—it means carving a new path you truly believe in. There’s a unique kind of perseverance and craft involved in that process, a combination of steady work and creative instinct that can’t be borrowed or replicated.

What Remains Even Without Success

Looking back, such stories sometimes turn into legends. Many attempts fail—they don’t catch on, or they collapse economically—but even then, the process itself remains fascinating. Invention is not a single result but a continuum of effort and curiosity. For those who can love that process, it is, without doubt, a kind of joy.

(富田勲さんと発明の精神)

最近、作曲の大家 - 富田勲さんのインタビューを見て、発明というものを「サイエンス」という言葉を使わずに語ってみたいなと思いました。富田勲は日本の電子音楽の祖と呼ばれており、そのお弟子さんである松武秀樹さんも同じくボブ・モーグのシンセサイザーを購入し、「第四のYMO」と呼ばれていたという話が知られています。

富田さんがモーグという電子音楽の機材をアメリカから取り寄せた理由が面白いと思いました。楽器というのは伝統や形式に守られすぎていて、イノベーションが少ない世界と彼はワグナー以前 vs 以降の楽曲を見て思ったそうです。楽器ごとに「こう鳴らすものだ」という定番があり、誰も疑わない。音の可能性がすでに出尽くしてしまったような閉塞感があったと述べています。

富田さんはその時点で既に音楽家として成功していましたが、自分の殻を破るために「自分で音を作れるらしい」という新しい挑戦を始めたのです。それが未知の領域を切り開くきっかけになりました。

(問題解決という創造の快楽)

もやっとした自分の悩みをどうにか解決できないかと考え続けることの尊さに通じます。文系とか理系に関わらず、こういった行為そのもののクリエイティブな側面が、発明や新しい試みをする人たちが好きなところなんじゃないかと思っています。

例えば、貧富の差がなくならないとか、難病が治らないといった問題に対しては、諦める方法を考えるのは簡単です。千個ぐらいの諦め方はすぐに思いつきます。これはクリエイティブではない方法です。反射的に出る思いつきのようなものでしょう。

(もちろん悩みの当事者からそういう言葉が直感的に出てしまうこと自体は否定しません)

一方、解きにくい問題を解決する方法は、理論上、百万通りでも十億通りでも存在しています。これが逆に解法、つまり発明のしにくさを表しています。

(実験と孤独の愉しみ)

百万通りの実験を個人で行うのは難しいことです。解決策は無限にあるのに、実験に使える時間は有限です。だからこそ、実験するという行為そのものが、限られた時間の中で光を見つけようとする探索になるんだと思います。よく「データを見ろ」「型にとらわれるな」などとスローガンのように言われますが、実際にはそんなに簡単な話ではないので。

富田さんがシンセサイザーの初期機材のダイヤルを手で回しながら音を探っていたように、真っ暗な闇の中で、頭を抱えながら少しずつ光を見つけていく孤独な作業こそが、本当の意味での創造的行為なのだと思います。そして終わってみれば、その悩みの時間そのものが、とても楽しい時間だったと感じるのです。

(理性的な仲間と反発する世界)

新しい試みに共感してくれる人たちは、たいてい理性的な人たちです。感情的なオーディエンスは、「そんなのやめておけ」「うまくいかない」と言うことが多い。「成功するには他人を真似した方がいい」と言う人もいる。でも、誰かの逆を行くためではなく、自分が信じる新しい道を切り開こうとするときに生まれる独特の創造的プロセスと地道な努力があると思っています。

(成功しなくても残るもの)

後になってみれば、それが美談になることもあります。ある人の試みがうまくいかず、流行らなかったり、経済的に失敗したりすることもある。けれども、たとえ消えていったとしても、そのプロセス自体が面白い。発明というのは結果ではなく過程の連続です。その過程を愛せる人にとって、それは間違いなく快感なのだと思っております。

(日本語下記に)

My Own Business Journey

Let me start with a bit about myself. I run a company that develops and sells enterprise applications. It’s been five years now, and the business has grown to a decent size. Of course, it hasn’t been easy—there have been countless struggles and setbacks. Still, I’ve kept going because there are moments when I truly understand the pain of my clients, especially the CEOs who run their own companies.

Understanding the Pain of Leaders

When you talk to CEOs, their top three headaches usually revolve around things like X, Y, and Z. Whenever I mention them, I often get a nod and a laugh: “Yes, exactly that!”
Their department heads face their own battles—compliance, sales pressure, advertising, and so on. When I say, “Your boss probably gives you a hard time about this, right?” it often hits the mark. Through these conversations, I’ve come to realize the essence of doing business: helping others and being compensated for genuinely useful work.

What You Can’t See from the Sidelines

You can attend all the seminars and read all the books you want, but they’ll never capture the atmosphere of the real world. There are so many unreported realities. No one writes blog posts about clashing with employees, and if they do, they’re probably not the kind of leaders worth learning from.
Reading that “many companies struggle with hiring” is not the same as sitting across from someone who can’t fill a position—or helping them fix it. The difference in depth is enormous. Only those who have been in the trenches can grasp the texture of that reality.

The Authenticity of Doing

I don’t mean to boast, but because I’m actually running a business, I can speak from real experience. I’d rather share words shaped by action and pain than repeat theories from books.
People who’ve just read an American management book often can’t resist giving advice. But being a practitioner, I know that kind of advice rarely matters. Because I’ve lived through it, I can pinpoint the real issue: “This is what’s really bothering you, isn’t it?”

The Growth That Comes from Helping Others

The skill of helping others doesn’t require payment to improve—it grows naturally.
Take someone who quits a big-name fashion house like Gucci or Louis Vuitton. Just managing stores or following Paris or Milan’s marketing playbook would get dull quickly. But starting your own small brand—printing notebooks or T-shirts on demand, building a community that blends fashion with music, film, or literature—now that’s exciting. You could even weave in your hometown or personal story. That, in itself, is already a creative act of value.

The Lessons Hidden in Failure

Even if you fail, it still matters. Small, human-scale experiments—however trivial they seem—become nourishment for growth.
I happen to be in the relatively glamorous IT industry, but the same applies to massage therapists, florists, T-shirt shop owners, or sushi chefs. Doing everything from start to finish, and studying where things go wrong, is far more valuable than smooth sailing.
It’s not just about the failure itself—it’s about how that experience allows you to speak with other entrepreneurs as equals. In that sense, the learning is a hundred, a thousand, even ten thousand times more valuable.

It may sound like a business essay, but it’s really about human pain and growth. Staying on the ground, acting, failing, and acting again—this is how we come to understand others’ struggles and how real creativity begins.

「成功よりも、うまくいかない経験が100倍役に立つ理由」

私自身の事業のこと

まず、私自身の話を少しします。現在、私たちは法人向けのアプリを自社で開発・販売しています。もう5年間続けており、そこそこの規模になっています。当然ながら、苦労も多く、想定通りに進まないことも山のようにあります。それでも続けているのは、この仕事を通して「お客さんの社長の痛みがわかる」瞬間があるからです。

経営者や事業部長の「痛み」を知る

社長が抱えるトップ3の悩みといえば、おそらく「X・Y・Z」のようなもので、実際に話してみると「そうそう、それそれ！」と共感が返ってくることが多いです。さらに、社長の下にいる事業部長クラスの人たちは、コンプライアンス、売上、広告など、また別の重たい課題を抱えています。
「社長にこういうことで怒られるんですよね？」と尋ねると、たいてい図星で、思わず笑ってしまうこともあります。こうしたリアルなやり取りを重ねるうちに、人の役に立ち、その対価としてお金をいただくという営みの本質を実感するようになりました。

現場にいないと見えない世界

セミナーを聞いたり本を読んだりしても、実際の現場の空気はなかなかわかりません。世の中には報道されない世界がたくさんあります。たとえば「従業員と揉めた」なんて話はどこにも書かれませんし、そんなことをブログに書いてしまうのはむしろダメな経営者でしょう。
採用で困っている会社が多い、という新聞記事を読むのと、実際に採用難に直面したり、それを解決する仕事をしたりするのとでは、話の深みがまったく違います。現場に身を置いた人間だけが知るリアリティがあるのです。

実行者としてのリアリティ

自慢するつもりはありませんが、私も事業を実際にやっているので、リアリティのある話ができます。上から目線で本に書いてあることをなぞるよりも、実行者としての「痛み」や「手触り」をもった言葉を伝えたいと思っています。
アメリカの大学教授が書いた本を読んだばかりの人は、それをアドバイスしたくて仕方ないようですが、私は「実行者」であるがゆえに、それだけでは意味がないと知っています。だからこそ、「あなたが困っているのはここですよね」と的確に指摘できるようになりました。

人を助ける力とスキルの磨かれ方

困っている人を助けるというスキルは、お金をもらわなくても磨かれていくものです。たとえば、今ファッション業界にいる人が、グッチやルイ・ヴィトンのような有名ブランドを辞めたとします。けれど、ただ店舗管理やフランス本社のマーケティング指示をなぞるだけでは、やはりつまらないでしょう。
オンデマンドでノートブックやTシャツを作って、自分のブランドを立ち上げる方法もあります。コミュニティを作って、音楽や映画、小説など文化的な発信を混ぜてもいい。自分の名前でやってもいいし、地元や思い入れのある場所をテーマにしてもいい。それ自体がすでに立派な挑戦です。

失敗の中にある学び

たとえそれが失敗しても、意味はあります。いろんな立場の人が人情味を持って小さな挑戦をすることは、一見ムダに見えても、実はすべてが自分の糧になります。
私はたまたまITという比較的華やかな業界にいますが、マッサージサロンでもTシャツ屋でも花屋でも寿司屋でも、何でも同じだと思います。1から10まで自分でやって、うまくいかないところを学ぶ経験こそが、本当に価値のあることです。
それは、失敗そのものよりも、さまざまな経営者と対等に話ができるという意味で、100倍も1000倍も、いや1万倍も役に立つ学びだと思っています。

このように書くと、単なる「ビジネスの話」に聞こえるかもしれませんが、実際には人間の痛みと成長の物語です。現場に立ち続け、実行を重ねることこそが、他者の痛みを理解し、支え合う力になる。その積み重ねが、やがて次の創造を生むのだと思います。