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# Quanta Magazine

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History is full of backstabbing rivalries: Edison and Tesla, Harding and Kerrigan, Tupac and Biggie. No less dramatic was a 16th-century conflict between Italian mathematicians Gerolamo Cardano, a brilliant but troubled polymath, and Niccolò Fontana, better known as Tartaglia (meaning “the stammerer,” after a teenage facial injury from a French soldier’s sword). The central issue: cubic equations.

Most high school students know how to solve quadratic equations, such as $latexx^2-x-3=0$, using the quadratic formula. It states that the solutions, or roots, of $latexax^2+bx+c=0$ are

$latexx=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

Are there similar formulas for equations of higher degree (having a larger power of x)? Determining this, in essence, was the task before Cardano, Tartaglia and their contemporaries.

Modern algebra as we know it — abstract symbolic expressions like those above and the familiar ways of manipulating them — dates to the 17th century, long after these scholars’ time. But algebraic thinking, and the ability to solve what we recognize as linear and quadratic equations, developed slowly over the preceding millennia.

In the 16th century, algebraic equations were still expressed rhetorically — in words, not symbols — and all coefficients had to be nonnegative, since mathematicians did not recognize negative numbers as legitimate. Without the notion of an unknown variable x, cubic equations of the form  $latexx^3+cx=d$ were instead described as “a cube and things equals a number,” and that was seen as different than “a cube equals things and a number,” $latexx^3=cx+d$. So while today we would view solving $latexax^3+bx^2+cx+d=0$ as one problem, at that time it was viewed as more than a dozen distinct problems, with the terms on one or the other side of the equal sign, or absent entirely.

Without modern symbolic algebra, mathematicians would reason geometrically. For instance, we can view the familiar expression $latex(a+b)^2=a^2+2ab+b^2$ as saying that the area of a square of side-length $latexa+b$  is the same as the sum of the areas of a square of side-length a, a square of side-length b, and two $latexa \times b$ rectangles.

Likewise, decomposing a cube with side-length t into six boxes shows that

(as long as $latext>u$).

Scipione del Ferro, a professor at the University of Bologna in the early 16th century, was first to make significant headway into solving cubic equations. Unfortunately, we do not know all his accomplishments, thanks to a curious culture of academic secrecy at the time. Rather than racing to publish their work and basking in the recognition of proving a theorem or solving a problem, scholars would challenge each other to “mathematical duels.” They would send each other challenging problems, and the one who solved the most was the winner. The victors often garnered professional advancement and more students. Thus, discoveries would sometimes be held in reserve, secret weapons to be deployed in future contests.

But we do know del Ferro could solve equations of the form $latexx^3+cx=d$, when c and d are positive. A cubic equation with no squared term, such as this one, is called a “depressed cubic.” Although no 16th-century mathematician would express it this way, del Ferro showed that one root is

This modern formula applies to any depressed cubic, but because cubic equations with different signs on the coefficients were considered different problems, del Ferro’s solution did not automatically carry over to other depressed cubics. We only know del Ferro could solve these cubics because he taught the technique to his student Antonio Fior, who bragged that he could solve such equations after del Ferro’s death.

Meanwhile, the self-taught Tartaglia discovered how to solve a different form of the cubic — one missing the linear term, cx. This set the stage for a mathematical duel between Fior and Tartaglia. In 1535, they exchanged 30 problems with a deadline of a month and a half. Tartaglia sent Fior a variety of problems, whereas the mathematically weaker Fior employed the “all eggs in one basket” strategy and sent Tartaglia 30 depressed cubics. Just days before the deadline, Tartaglia figured out how to solve them, and in two hours finished all 30. Fior, meanwhile, solved none of his problems. News spread throughout Italy of Tartaglia’s achievement, and Fior, humiliated, faded from view.

The prevailing wisdom was that solving the cubic was impossible, so Tartaglia’s accomplishment shocked Cardano. At this time, Cardano was a highly sought physician but cantankerous, beset by trouble after trouble. He gambled, he struggled with misbehaving sons, he was jailed during the Inquisition and more. Yet he ended up making contributions in mathematics, medicine, philosophy, religion, music and physics. Decades later, Gottfried Leibniz wrote, “Cardano was a great man with all his faults; without them, he would have been incomparable.” His collected works fill 7,000 pages and include the first serious investigations of probability theory.

Cardano tried, and failed, to replicate Tartaglia’s success with the cubic, so he began a pressure campaign to convince Tartaglia to share his method, even promising a vow of secrecy:

I swear to you by the Sacred Gospel, and on my faith as a gentleman, not only never to publish your discoveries, if you tell them to me, but I also promise and pledge my faith as a true Christian to put them down in cipher so that after my death no one shall be able to understand them.

Eventually, in 1539, Tartaglia relented and shared his technique for depressed cubics with Cardano, but he did not share the proof that it worked. For the clever Cardano, however, just knowing the method was enough to discover the underlying mathematics. Before long, Cardano could solve any depressed cubic. He then observed that substituting

$latexx=t-\frac{b}{3a}$

into $latexax^3+bx^2+cx+d=0$ yields a depressed cubic with variable t. By solving this equation for t and plugging that back into the substitution formula, he could find x. Thus, Cardano was able to solve every cubic equation.

Despite his vow to Tartaglia, Cardano taught these results to his talented assistant Ludovico Ferrari. Although he began as Cardano’s servant, Ferrari eventually became Cardano’s mathematical equal. By helping Cardano with his work on the cubic, he became so algebraically adept that he discovered how to reduce any quartic equation (one of degree 4) to a cubic. Thus, Cardano and Ferrari could solve any equation of degree four or less.

Cardano recognized the importance of these accomplishments and desperately wanted to publish the results. But since they all grew from the seeds planted by Tartaglia, doing so would break his oath.

Then, on a trip to Bologna in 1543, Cardano saw in del Ferro’s notebooks that he had solved the depressed cubic before Tartaglia. In Cardano’s mind, this discovery freed him of his obligation to Tartaglia. Two years later, Cardano published Ars Magna (Great Art), which contained his and Ferrari’s work on cubic and quartic equations.

Tartaglia was livid, even though Cardano acknowledged his work in the book. Tartaglia accused Cardano of theft and of breaking a sacred vow. Cardano left the rebukes to his loyal attack dog, Ferrari. The acrimonious back-and-forth, in the form of public pamphlets, continued for many months, leading to a mathematical duel between Tartaglia and Ferrari and eventually a public debate in Ferrari’s hometown, Milan. Tartaglia would much rather have battled the esteemed Cardano, but Cardano refused. Details are scarce, but the debate went terribly for Tartaglia, especially with the raucous hometown crowd. The next day, when it was time to continue the debate, Tartaglia was nowhere to be found — he’d left Milan.

Ferrari was flooded with job offers, and Tartaglia’s reputation was ruined. Despite many notable accomplishments beyond those related to the cubic, Tartaglia died penniless and largely unknown, whereas Cardano achieved everlasting fame. Many argue that the publication of Ars Magna marked the beginning of modern mathematics.

Having conquered cubic and quartic equations, mathematicians wondered how high they could go. As it turned out, not very far.

The story of quintic equations (polynomials of degree 5) is also fascinating, and it has a shocking conclusion: In general, it is impossible to express the roots of $latexax^5+bx^4+cx^3+dx^2+ex+f=0$ in terms of a, b, c, d, e and f using only addition, subtraction, multiplication, division and nth roots. For instance, the polynomial $latexx^5-x+1$ has a root of approximately $latex–1.67304$, but the exact value is impossible to express with those tools.

Niels Abel gave the first full proof of this fact in 1824, nearly three centuries after Ars Magna. Then, in 1830, the 18-year-old political firebrand Évariste Galois extended this work by giving exact criteria for when a polynomial of any degree is solvable. Although Galois died two years later in a duel (one fought with guns, not mathematics), his contributions to mathematics were oversized.

These impossibility results were not the end of the story. Mathematicians still study polynomials, their roots and their properties. As but one example, a famous problem proposed by David Hilbert in 1900 is about the roots of seventh-degree polynomials. It was thought to have been resolved in the 1950s but is now the subject of renewed interest. Presumably, modern mathematicians can make headway on the problem without re-creating the rivalry surrounding the cubic.

# PrivaCrip to Power a Blockchain Network That Could Follow Cardano & XRP

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When the first decentralised blockchain was popularised in 2008 by Satoshi Nakamoto, the door was open for people across the globe to transact more securely and seamlessly. Many blockchain networks have been built and powered by native crypto tokens, including PrivaCrip (PRCR).

Native tokens exist for several reasons. Aside from acting as digital currencies, they perform various functions. Helping blockchains improve data traceability, security, transparency, and trustworthiness shared throughout the network. They also assist in reducing costs through their ever-evolving efficient methods.

Cardano (ADA) and Ripple (XRP) are among the best blockchain-based cryptos that this article will discuss. We will also look at how PrivaCrip (PRCR) will power a blockchain network set to launch into the crypto space.

## Top Tier Crypto Makes its Mark – Cardano (ADA)

Cardano (ADA) has been around in the crypto space since 2017. The Proof-of-Stake (PoS), Ouroborous Cardano (ADA), was established using a research-based methodology.

Research is the backbone of Cardano (ADA) and is integral to its success in the cryptocurrency market. Cardano (ADA) uses an evidence-driven mechanism and peer-reviewed research to run its ecosystem to drive its evolution.

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Among many things, Cardano (ADA) helps establish DeFi products and provides a suite of financial solutions. Crypto enthusiasts can stake Cardano (ADA) for a chance to earn rewards for their holdings.

With 2022 providing challenges in cryptocurrencies across the globe, Cardano (ADA) is one of the few cryptos to see a surge in on-chain trading volume this year. CoinMarketCap lists Cardano (ADA) as the 7th largest crypto by market capitalization at the time of writing.

## Making Waves – Ripple (XRP)

Launched in 2012, Ripple (XRP) is a leading provider of crypto solutions for businesses. Ripple (XRP) provides developers with a solid open-source foundation for executing demanding projects. It is fast, energy-efficient, cost-effective, and reliable.

The solutions provided by Ripple (XRP) enable developers to solve inefficiencies, including asset tokenization and remittance. Therefore, individuals and businesses may apply Ripple (XRP) for DeFi, payments, tokenization, and more.

Crypto enthusiasts with a long-term focus may find Ripple (XRP) a valuable investing tool. Token holders can use Ripple (XRP) for staking its mainnet platform for a chance to grow their earnings and maximise profit.

## The New Face Of Privacy – PrivaCrip (PRCR)

The soon-to-launch PrivaCrip (PRCR) is a cryptocurrency built for Web3 data privacy. It will allow users to develop and use permissionless and privacy-preserving applications. This unique feature of PrivaCrip (PRCR) secures apps, protects users, and fosters the creation of many new Web3 uses.

PrivaCrip (PRCR) will drive a blockchain that creates a more empowering and inclusive internet. It will enable developers to build secure dApps using private contracts based on original privacy research (OPR) to revolutionise Web3.

PrivaCrip (PRCR) will begin pre-sale soon and has an initial token price of 0.10 USD. According to the whitepaper, there will be a total supply of 190 million tokens, of which 40% of the total supply will go to the pre-sale.

Crypto enthusiasts who purchase PrivaCrip (PRCR) during the pre-sale will receive the most significant long-term rewards and bonuses. Those who buy with Bitcoin (BTC) will receive a 10% bonus. Also, those using Ethereum (ETH) to purchase will receive 15% of the total PrivaCrip (PRCR) they purchase as a bonus.

10% of PrivaCrip’s (PRCR) total supply is up for distribution as staking rewards, meaning that token holders who stake will receive rewards for their contribution to the crypto’s advancement.

## The Bottomline

Blockchain networks are continuously evolving, and so are the crypto tokens that power them. Cardano (ADA) and Ripple (XRP) have contributed tremendously to the success of their blockchain networks, gathering mass appeal.

PrivaCrip (PRCR), for its part, will drive a security-enabled blockchain and scalability in private DeFi, empower data tokenization, and bring together a fast-growing community. Acting on its promise, it has the potential to become the next big mover in cryptocurrency.

For more on PrivaCrip (PRCR), check below:

Website: http://privacrip.io/

Telegram: https://t.me/PrivaCripOfficial

# Cardano (ADA) Withdrawal Fees Waved by Bitrue Exchange

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Alex Dovbnya

Singapore-based cryptocurrency exchange made Cardano one of its base trading pairs earlier this year

Singapore-based cryptocurrency exchange Bitrue announced that it had temporarily waived withdrawal fees for the Cardano (ADA) cryptocurrency.

The trading platform says that the move is meant to celebrate the support it received from the community behind the cryptocurrency.

Users will be able to withdraw ADA without paying any fees until mid-September.

Bitrue has stressed that it wants users to have “as much choice as possible” when it comes to custody options. Those who want to hold their ADA tokens will not be able to do so without paying an additional commission.

At the same time, Bitrue has touted its “Power Piggy” yield-farm investment program for those who want to earn passive income with ADA.

As reported by U.Today, Bitrue introduced the token as its base currency back in February.

In June 2021, it also became the first cryptocurrency trading platform to add support for Cardano-based native tokens.

Last month, Bitrue also announced a staking initiative with the ADA cryptocurrency. It decided to contribute a million tokens to some of the top Cardano staking pools in an effort to boost the level of decentralization.

Earlier this year, SundaeSwap (SUNDAE), the native token of the SundaeSwap exchange, also became available on Bitrue.

# WingRiders Losses 59% Of Its Total Value Locked, Plunges Cardano TVL Below $100M Published on – Advertisement – Popular Cardano-based decentralized exchange WingRiders has lost its position as the most valuable DEX on Cardano. WingRiders slumped from the first position to third after losing more than 59% of the total value locked (TVL) on the platform in the past 30 days. At the time of writing this line, WingRiders is now the third-largest DEX by total value locked, with a TVL of$19.66 million. It is noteworthy that WingRiders maintained the top spot for more than two months, claiming  42% of TVL On Cardano. However, the DEX has fallen from glory ever since.

Minswap Is Now Cardano’s Most Valuable DEX

The development also affected the total value locked on Cardano. The total value locked on Cardano has fallen below $100 million in the last 30 days after the WingRiders TVL slumped. Interestingly, Minswap has taken the lead to become the most valuable decentralized exchange on Cardano after the massive decline of WingRiders’ TVL. At press time, Minswap is leading, with a$49.09 million total value locked on the platform. The DEX hit the milestone after recording a massive growth of 30.9% in its TVL in the past 30 days. Minswap accounts for 51.82% of the total value locked on Cardano.

SundaeSwap, the first decentralized exchange on Cardano, is now the second most valuable DEX in the ecosystem. SundaeSwap is second in the ranking, with $19.72 million in TVL. Like WingRiders, SundaeSwap has also lost a significant amount of its TVL over the past 30 days. In the past month, SundaeSwap has lost nearly 10% of the funds locked on the platform. Commenting on the development, Cardano Daily, a platform that shares new developments about Cardano, said: “After the major decline of @wingriderscom, the TVL scenario of the Cardano ecosystem has stabilized back to normal as we have seen in the past few weeks. @MinswapDEX still led the ecosystem with$51M TVL, contributing 50% of the ecosystem.”