ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate this sentence to Genoese:
For example, Dedekind was the first to prove Fermat's two-square theorem using the unique factorization of Gaussian integers.
The Ligurian (Genoese) translation is:
Pe fâ un exempio, Dedekind o l’é stæto o primmo à demostrâ o teorema di doî quaddræ de Fermat pe mezo da fattorizzaçion unica di intreghi gaussien.
Translate to Ligurian:
Other applications of Euclid's algorithm were developed in the 19th century.
The Ligurian (Genoese) translation is:
De atre applicaçioin à l’algoritmo d’Euclide en stæte sviluppæ do secolo XIX.
Translate from English to Ligurian (Genoese dialect):
Several novel integer relation algorithms have been developed, such as the algorithm of Helaman Ferguson and R.W. Forcade (1979) and the LLL algorithm.
The Ligurian (Genoese) translation is:
L’é stæto sviluppou varri neuvi algoritmi de relaçioin intreghe, comme l’algoritmo de Helaman Ferguson e R.W. Forcade (1979) e l’algoritmo LLL.
Translate from English to Ligurian:
The players take turns removing m multiples of the smaller pile from the larger.
The Ligurian (Genoese) translation is:
I zugoei, un dòppo l’atro, remeuvan m moltipli da-a pilla ciù picciña à quella ciù grande.
What’s the Ligurian translation of this sentence?
By allowing u to vary over all possible integers, an infinite family of solutions can be generated from a single solution (x1, y1).
The Ligurian (Genoese) translation is:
De permette à u de cangiâ in sce tutti i intreghi poscibili, se peu generâ unna famiggia infinia de soluçioin da unna soluçion sencia (x1, y1).
Translate this sentence to Genoese:
In this field, the results of any mathematical operation (addition, subtraction, multiplication, or division) is reduced modulo 13; that is, multiples of 13 are added or subtracted until the result is brought within the range 0–12.
The Ligurian (Genoese) translation is:
Inte sto campo, i resultati de unn’operaçion matematica chesesegge (addiçion, sottraçion, moltiplicaçion e divixon) vëgnan reduti a-o mòdolo 13; saieiva à dî che i moltipli de 13 vëgnan azzonti ò sottræti fintanto che o resultato o no l’intra inte l’intervallo 0-12.
Translate to Ligurian (Genoese):
Now assume that the result holds for all values of N up to M − 1.
The Ligurian (Genoese) translation is:
Oua supponemmo che o resultato o vagge pe tutti i valoî de N fin à M - 1.
What’s the Genoese translation of this sentence?
For illustration, the probability of a quotient of 1, 2, 3, or 4 is roughly 41.5%, 17.0%, 9.3%, and 5.9%, respectively.
The Ligurian (Genoese) translation is:
Pe fâ un exempio, a probabilitæ de un quoçiente de 1, 2, 3 ò 4 o l’é approscimativamente 41,5%, 17,0%, 9,3% e 5,9%, respettivamente.
Can you translate this text to Ligurian?
One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor.
The Ligurian (Genoese) translation is:
Unna metodologia inefficaçe pe attrovâ o MDC de doî numeri naturali a e b o l’é de carcolâ tutti i seu divisoî commun; o MCD o l’é donca o ciù grande divisô commun.
Translate this sentence to Ligurian:
As noted above, the GCD equals the product of the prime factors shared by the two numbers a and b. Present methods for prime factorization are also inefficient; many modern cryptography systems even rely on that inefficiency.
The Ligurian (Genoese) translation is:
Comme l’é stæto fæto nottâ chì de d’ato, o MDC o l’é pægio do produto di fattoî primmi scompartii da-i doî numeri primmi a e b. I metodi che se deuvia ancheu pe-a fattorizzaçion di primmi son ascì ineffifaçi; tanti scistemi de crittografia moderni arrivan fiña à basâse in sce st’inneficaçitæ.
Translate from English to Genoese:
Lehmer's GCD algorithm uses the same general principle as the binary algorithm to speed up GCD computations in arbitrary bases.
The Ligurian (Genoese) translation is:
L’algoritmo MCD de Lehmer o deuvia o mæximo prinçipio generale de l’algoritmo binäio pe accelerâ i carcoli MCD in sce base arbiträie.
Translate this sentence to Genoese:
The Euclidean algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can also be defined.
The Ligurian (Genoese) translation is:
L’algoritmo euclideo o peu vegnî deuviou pe resòlve de equaçioin lineare diofantiñe e problemi de resto cineise pe polinòmmi; se peuan definî ascì fraçioin continuæ de polinòmmi.
What’s the Ligurian translation of this sentence?
Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.
The Ligurian (Genoese) translation is:
Tutti i dominni euclidëi en di dominni de fattorizzaçion unichi (UFD), sciben che no l’é veo o conträio.
Translate from English to Genoese:
A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal.
The Ligurian (Genoese) translation is:
Un dominio euclideo o l’é sempre un dominio ideale prinçipâ (PID), un dominio integrale donde tutti i ideali en di ideali prinçipæ.
Translate to Ligurian (Genoese):
Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
The Ligurian (Genoese) translation is:
I numeratoî e denominatoî en deuviæ ascì inte de fraçioin no commun, compreiso e fraçioin compòste, quelle complesse e i numerali mescci.
Translate from English to Ligurian:
The term was originally used to distinguish this type of fraction from the sexagesimal fraction used in astronomy.
The Ligurian (Genoese) translation is:
A-o comenso o termine o l’ea deuviou pe distingue sto tipo de fraçion da-a fraçion secsageximale deuviæ inte l’astronomia.
Translate to Genoese:
This was explained in the 17th century textbook The Ground of Arts.
The Ligurian (Genoese) translation is:
Sto fæto o s’attreuva spiegou into libbro de testo The Ground of Arts, do secolo XVII.
Translate from English to Genoese:
The product of a fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction.
The Ligurian (Genoese) translation is:
O produto de unna fraçion e o seu reçiproco o l’é 1, donca o reçiproco o l’é l’inverso moltiplicativo de unna fraçion.
What’s the Ligurian translation of this sentence?
The remainder becomes the numerator of the fractional part.
The Ligurian (Genoese) translation is:
O resto o vëgne o numeratô da parte fraçionäia.
What’s the Genoese translation of this sentence?
Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is .
The Ligurian (Genoese) translation is:
Da-o momento che 5×17 (= 85) o l’é ciù grande de 4×18 (= 72), o resultato da comparaçio o l’é .
Translate to Ligurian (Genoese):
Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth.
The Ligurian (Genoese) translation is:
Dæto che a tersa parte de unna quarta parte a l’é unna dozzeña parte, doe terse parte de unna quarta parte en doe dozzeñe parte.
Translate to Ligurian (Genoese):
Sometimes an infinite repeating decimal is required to reach the same precision.
The Ligurian (Genoese) translation is:
De vòtte l’é neçessäio un deximâ repetuo à l’infinio pe razzonze a mæxima preçixon.
Translate from English to Ligurian:
The Egyptians used Egyptian fractions BC.
The Ligurian (Genoese) translation is:
I egiçien deuviavan e fraçioin egiçie a.C.
Translate this sentence to Genoese:
Their methods gave the same answer as modern methods.
The Ligurian (Genoese) translation is:
I seu metodi davan a mæxima respòsta che quelli moderni.
Translate to Ligurian:
A modern expression of fractions known as bhinnarasi seems to have originated in India in the work of Aryabhatta, Brahmagupta, and Bhaskara.
The Ligurian (Genoese) translation is:
Unn’esprescion moderna de fraçioin, conosciua comme bhinnarasi, a pâ ëse nasciua inte l’India pe mezo de euvie de Aryabhatta, Brahmagupta e Bhaskara.
Translate from English to Ligurian (Genoese dialect):
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
The Ligurian (Genoese) translation is:
Inta matematica, l’aritmetica modulare a l’é un scistema d’aritmetica pe intreghi, donde i numeri «s’ingheuggian» quand’arrivan à un çerto valô, ciammou mòdolo.
What’s the Genoese translation of this sentence?
A very practical application is to calculate checksums within serial number identifiers.
The Ligurian (Genoese) translation is:
Unn’applicaçion assæ prattica a l’é o carcolo de checksum drento di identificatoî de numeri seriali.
Translate to Ligurian:
RSA and Diffie–Hellman use modular exponentiation.
The Ligurian (Genoese) translation is:
RSA e Diffie-Hellman deuvian l’esponençiaçion modulare.
Translate this sentence to Genoese:
It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the integers and the rational numbers.
The Ligurian (Genoese) translation is:
O l’é deuviou inte implementaçioin ciù efficaçe do MDC polinomiale, de l’algebra lineare esatta e di algoritmi de base de Gröbner in scî intreghi e in scî numeri raçionæ.
Translate this sentence to Genoese:
The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context.
The Ligurian (Genoese) translation is:
L’operaçion mòdolo, implementâ inte tanti lenguaggi de programmaçioin e carcolatoî, a l’é unn’applicaçion de l’aritmetica modulare che de spesso a ven deuviâ inte sto contesto.
Can you translate this text to Ligurian?
The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand.
The Ligurian (Genoese) translation is:
O metodo do tio di neuve o l’òffre un contròllo spedio di carcoli deximæ d’aritmetica fæti à man.
Translate to Ligurian (Genoese):
A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem.
The Ligurian (Genoese) translation is:
Un scistema lineare de congruense o peu vegnî soluçionou inte un tempo polinomiale con unna forma d’eliminaçio gaussiaña; pe di detaggi, s’ammie teorema de congruensa lineare.
Translate from English to Genoese:
The multiplication of integers (including negative numbers), rational numbers (fractions) and real numbers is defined by a systematic generalization of this basic definition.
The Ligurian (Genoese) translation is:
A moltiplicaçion de numeri intreghi (compreiso quelli negativi), numeri raçionæ (fraçioin) e numeri reali a l’é definia da unna generalizzaçion scistematica de sta definiçion de base.
Translate from English to Ligurian:
The product of two measurements is a new type of measurement.
The Ligurian (Genoese) translation is:
O produto de doe mesuaçioin o l’é un neuvo tipo de mesuaçion.
Translate to Ligurian (Genoese):
The inverse operation of multiplication is division.
The Ligurian (Genoese) translation is:
L’operaçion inversa da moltiplicaçion a l’é a divixon.
Translate from English to Genoese:
The division of a number other than 0 by itself equals 1.
The Ligurian (Genoese) translation is:
Se un numero despægio da-o zero o vëgne diviso pe lê mæximo, o resultato o l’é 1.
Translate this sentence to Ligurian:
This implicit usage of multiplication can cause ambiguity when the concatenated variables happen to match the name of another variable, when a variable name in front of a parenthesis can be confused with a function name, or in the correct determination of the order of operations.
The Ligurian (Genoese) translation is:
Quest’utilizzaçion impliçita da moltiplicaçion a peu provocâ de ambiguitæ quande capita che e variabile concadenæ correspondan a-o nomme de unn’atra variabile, quande un nomme de variabile davanti à unna parentexi o peu vegnî confuso co-o nomme de unna fonçion ò inta corretta determinaçion de l’ordine de operaçioin.
Translate from English to Genoese:
The numbers to be multiplied are generally called the "factors".
The Ligurian (Genoese) translation is:
Pe commun i numeri da moltiplicâ en ciammæ «fattoî».
What’s the Genoese translation of this sentence?
Also as the result of a multiplication does not depend on the order of the factors, the distinction between "multiplicand" and "multiplier" is useful only at a very elementary level and in some multiplication algorithms, such as the long multiplication.
The Ligurian (Genoese) translation is:
Ascì pe-o fæto che o resultato de unna moltiplicaçion o no depende da l’ordine di fattoî, a distinçion fra «moltiplicando» e «moltiplicatô» a serve solo che pe un livello assæ elementare e inte çerti algoritmi de moltiplicaçion, comme a moltiplicaçion longa.
Translate to Ligurian:
The result of a multiplication is called a product.
The Ligurian (Genoese) translation is:
O resultato de unna moltiplicaçion o l’é dito o produto.
Translate this sentence to Ligurian:
The slide rule allowed numbers to be quickly multiplied to about three places of accuracy.
The Ligurian (Genoese) translation is:
O regolo carcolatô o permetteiva de carcolâ a-a spedia i numeri con trei ponti de preçixon pöcassæ.
What’s the Ligurian translation of this sentence?
The complex numbers do not have an ordering.
The Ligurian (Genoese) translation is:
I numeri complesci no gh’an unna sequensa.
Translate this sentence to Genoese:
Here we have identity 1, as opposed to groups under addition where the identity is typically 0.
The Ligurian (Genoese) translation is:
Chì emmo l’identitæ 1, a-o conträio di gruppi sotta addiçion donde l’identitæ a l’é tipicamente 0.
What’s the Ligurian translation of this sentence?
To see this, consider the set of invertible square matrices of a given dimension over a given field.
The Ligurian (Genoese) translation is:
Comme demostraçion, piggiæ l’insemme de matrixe quaddræ che se peuan invertî de unna dimenscion determinâ in sce un campo determinou.
What’s the Ligurian translation of this sentence?
Another fact worth noticing is that the integers under multiplication is not a group—even if we exclude zero.
The Ligurian (Genoese) translation is:
Un atro fæto ch’o va nottou o l’é che i intreghi sotta moltiplicaçion no son un gruppo, fiña se se lascia feua o zero.
Translate from English to Ligurian:
In mathematics, a percentage (from Latin per centum "by a hundred") is a number or ratio expressed as a fraction of 100.
The Ligurian (Genoese) translation is:
Inta matematica unna perçentuale (da-o latin per centum) o l’é un numero ò un rappòrto espresso con unna fraçion in sce 100.
Translate to Ligurian:
Computation with these fractions was equivalent to computing percentages.
The Ligurian (Genoese) translation is:
O carcolo con ste fraçioin o l’ea equivalente a-o carcolo de perçentuale.
Translate to Genoese:
Whenever communicating about a percentage, it is important to specify what it is relative to (i.e., what is the total that corresponds to 100%).
The Ligurian (Genoese) translation is:
Quande se communica unna perçentuale, l’é importante speçificâ à cös’a l’é ligâ (ò sæ quæ o l’é o totale ch’o corresponde a-o 100%).
Can you translate this text to Ligurian?
When speaking of a "10% rise" or a "10% fall" in a quantity, the usual interpretation is that this is relative to the initial value of that quantity.
The Ligurian (Genoese) translation is:
Quande se parla de un «10% d’aumento» ò de un «10% de diminuçion» de unna çerta quantitæ, l’interpretaçion commun a l’e che questi seggian ligæ a-o valô de comenso de quella quantitæ.
Translate this sentence to Ligurian:
The term has been attributed to Latin per centum.
The Ligurian (Genoese) translation is:
O termine o ne vëgne da-o latin per centum.
Translate to Ligurian (Genoese):
Grammar and style guides often differ as to how percentages are to be written.
The Ligurian (Genoese) translation is:
E guidde de grammatica e de stile tante vòtte dan de indicaçioin despæge in sce comme s’à da scrive e perçentuale.
Translate to Ligurian (Genoese):
When interest rates are very low, the number 0 is included if the interest rate is less than 1%, e.g. "% Treasury Stock", not "% Treasury Stock".)
The Ligurian (Genoese) translation is:
Quande i tasci d’interesse en assæ basci, ven incluso o numero 0 se o tasso d’interesse o l’é ciù basso de l’1%, comme «% Treasury Stock», no «% Treasury Stock»).
Translate to Ligurian (Genoese):
Subtraction also obeys predictable rules concerning related operations, such as addition and multiplication.
The Ligurian (Genoese) translation is:
A sottraçion ascì a l’é ligâ à de regole preveddibili ch’an da fâ con de operaçioin correlæ, comme l’addiçion e a moltiplicaçion.
What’s the Ligurian translation of this sentence?
Performing subtraction on natural numbers is one of the simplest numerical tasks.
The Ligurian (Genoese) translation is:
Realizzâ a sottraçion in scî numeri naturali o l’é un di compiti ciù senci da fâ co-i numeri.
Translate from English to Ligurian:
Subtraction" is an English word derived from the Latin verb subtrahere, which in turn is a compound of sub "from under" and trahere "to pull".
The Ligurian (Genoese) translation is:
«Sottraçion» a l’é unna paròlla ch’a ne ven da-o latin subtrahere, compòsta à seu vòtta da sub «de sotta» e trahere «levâ».
Can you translate this text to Ligurian?
From position 3, it takes no steps to the left to stay at 3, so .
The Ligurian (Genoese) translation is:
Da-a poxiçion 3, no gh’é beseugno de fâ di pasci a-a manciña pe arrestâ in sciô 3, donca .
What’s the Genoese translation of this sentence?
To represent such an operation, the line must be extended.
The Ligurian (Genoese) translation is:
Pe representâ st’operaçion, a linea a dev’ëse esteisa.
Translate from English to Ligurian (Genoese dialect):
The leading digit "1" of the result is then discarded.
The Ligurian (Genoese) translation is:
Dapeu a primma giffra 1 do resultato a ven descartâ.
Translate this sentence to Genoese:
In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place.
The Ligurian (Genoese) translation is:
Into pòsto de dëxeñe 0 o l’é ciù basso che 1, donca o 0 o ven aumentou de 10 e a differensa con 1, ch’a l’é 9, a ven scrita into pòsto de dëxeñe.
What’s the Genoese translation of this sentence?
The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place.
The Ligurian (Genoese) translation is:
A sottraçion a se mescia dapeu into pòsto de çentanæa, donde o 6 o no l’é menô de 5, donca a differensa ven scrita into pòsto de çentanæa do resultato.
Translate to Ligurian:
Rather it increases the subtrahend hundred's digit by one.
The Ligurian (Genoese) translation is:
Ciutòsto o l’aumenta de un a giffra do sottraendo çento.
Translate this sentence to Ligurian:
The answer is 1, and is written down in the result's hundred's place.
The Ligurian (Genoese) translation is:
A respòsta a l’é 1, e a se marca into pòsto de çentanæe into resultato.
Translate from English to Ligurian (Genoese dialect):
This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica, where he claimed that he had a proof that was too large to fit in the margin.
The Ligurian (Genoese) translation is:
Sto teorema o l’é stæto ipotizzou pe-a primma vòtta da Pierre de Fermat do 1637 in sce l’oexin de unna pagina do volumme Arithmetica, donde o sostegniva ch’o l’aiva unna preuva tròppo grande pe vegnî representâ inte quello spaçio coscì piccin.
Translate this sentence to Ligurian:
It was the first major theorem to be proved using a computer.
The Ligurian (Genoese) translation is:
O l’é stæto o primmo teorema de unna çerta relevansa à vegnî demostrou pe mezo de un computer.
Translate this sentence to Ligurian:
Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps.
The Ligurian (Genoese) translation is:
Pe de ciù, tutte e mappe che porrieivan ëse, potençialmente, un contraexempio devan avei unna parte ch’a s’assomeggia à uña de ste 1936 mappe.
Translate to Ligurian:
It was originally formulated in 1908, by Steinitz and Tietze.
The Ligurian (Genoese) translation is:
O l’é stæto formulou pe-a primma vòtta do 1908 da Steinitz e Tietze.
Translate from English to Ligurian:
A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with qk elements containing that field.
The Ligurian (Genoese) translation is:
Unna varietæ V in sce un campo finio con q elementi o l’à un numero finio de ponti raçionæ, coscì comme di ponti in sce ògni campo finio con qk elementi contegnui inte quello campo.
Translate from English to Ligurian (Genoese dialect):
Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold).
The Ligurian (Genoese) translation is:
Stæto ipotizzou pe-o primmo da Henri Poincaré, o teorema o l’à da fâ con un spaçio ch’o s’assomeggia a-o spaçio à træ dimenscioin ordinäio ma ch’o l’é connesso, de dimenscioin finio e sensa confin (un 3-manifold serrou).
Translate from English to Ligurian (Genoese dialect):
After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.
The Ligurian (Genoese) translation is:
Squæxi çent’anni dòppo i sfòrsci di studioxi da matematica, Grigori Perelman o l’à presentou a preuva de st’ipòtexi inte trei articoli presentæ do 2002 e do 2003 in sce arXiv.
Translate to Ligurian (Genoese):
Perelman completed this portion of the proof.
The Ligurian (Genoese) translation is:
Perelman o l’à completou questa porçion da preuva.
Translate to Ligurian (Genoese):
Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjectured that the answer is no.
The Ligurian (Genoese) translation is:
À un mòddo informale, o domanda se pe tutti i problemi che peuan ëse verificæ a-a spedia da un computer pòsse ëse soluçionæ a-a spedia ascì da-o mæximo computer; pe-a ciù parte s’ipotizza che a respòsta a segge no.
Translate from English to Ligurian:
It has not been proven which one is false, but it is widely believed that the first conjecture is true and the second one is false.
The Ligurian (Genoese) translation is:
No l’é stæto provou quæ o l’é o fäso, ma pe commun se pensa che a primma ipòtexi a segge vea e a segonda a segge fäsa.
Translate from English to Ligurian:
For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (over a trillion).
The Ligurian (Genoese) translation is:
Pe dâ un exempio, l’ipòtexi de Collatz, ligâ a-o fæto se çerte sequense d’intreghi aggian da fin ò no, a l’é stæta testâ pe tutti i intreghi fin à 1,2 × 1012 (ciù de mille miliardi).
Translate to Ligurian (Genoese):
That evidence may be of various kinds, such as verification of consequences of it or strong interconnections with known results.
The Ligurian (Genoese) translation is:
Quelle preuve peuan ëse de diversa qualitæ, comme a verifica de seu conseguense ò de fòrte interconnescioin con di resultati conosciui.
Translate this sentence to Ligurian:
One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as "brute force": in this approach, all possible cases are considered and shown not to give counterexamples.
The Ligurian (Genoese) translation is:
Un metodo de demostraçion, ch’o se peu applicâ quand’existe solo che unna quantitæ finia de caxi che porrieivan portâ à di contraexempi, o l’é conosciume comme «fòrsa bruta»; inte sta metodologia se considera tutti i caxi poscibili e se demostra che n’an de contraexempi.