Number Theory

Original Titles
English Titles

26Observations on a theory of Fermat and others on looking at prime numbers
29On the solution of a problem of Diophantus
36Solution of problems of arithmetic of finding numbers which, when divided by given numbers, leave given remainders
54A proof of certain theorems regarding prime numbers
98The proofs of some arithmetic theorems
100On Amicable Numbers
134Theorems on divisors of numbers
152On amicable numbers
158Various analytical observations about combinations
164Theorems about the divisors of numbers contained in the form paa ± qbb
167On the solution of a most difficult problem proposed by Fermat
175Discovery of an extraordinary law of numbers in relation to the sum of their divisors
191On the partitions of numbers
228On numbers which are the sum of two squares
241Proof of a theorem of Fermat that every prime number of the form 4n+1 is the sum of two squares
242Proof of a theorem of Fermat that every number whether whole or fraction is the sum of four or fewer squares
243Observations on the sums of divisors
244A demonstration of a theorem on the order observed in the sums of divisors
253On indeterminate problems which appear to be quite determinate
255General solution of certain Diophantine problems, which are ordinarily thought to admit only special solutions
256Example of the use of observation in pure mathematics
262Theorems about the remainders left by division by powers
270The solution of a problem about searching for three numbers, of which the sum and not only their product but the sum of their products two apiece, are square numbers
271Demonstration of a new method in the Theory of Arithmetic
272A reinforcement of some arithmetic theorems, supported by several demonstrations
279On the resolution of formulas of squares of indeterminates by integral numbers
283On very large prime numbers
323Concerning the use of a new algorithm to solve the Pell problem
369How very large numbers are to be tested for whether they are prime or not
394On the partition of numbers into a number of parts of a given type
405A solution of a problem about two numbers which are searched for, of which their product, increased or decreased by their sum or difference, will be a square
427An investigation of a certain Diophantine problem
428Observations about two biquadratics, of which the sum is able to be resolved into two other biquadratics
445Proof that every integer is the sum of four squares
449Demonstrations about the residues resulting from the division of powers by prime numbers
451A solution of the problem of finding a triangle, in which the lines from each angle bisecting the opposite sides are rational
452The resolution of the equation Ax2 + 2Bxy + Cy2 + 2Dx + 2Ey + F = 0 by rational and integral numbers
454On the resolution of irrationals by continued fractions, where a certain minor new and singular type is set forth
461Extract of a letter by Mr. Euler to Mr. Bernoulli father concerning the memoire published by them in 1771, p. 318
466A singular Diophantine problem
467On the table of prime numbers continued up to one million and beyond, in which at once all the non-prime numbers are expressed by their smallest divisors
474A solution of several Diophantine problems
498Extract of a letter from Mr. Euler to Mr. Beguelin from May 1778
515De casibus quibusdam maxime memorabilibus in analysi indeterminata, ubi imprimis insignis usus calculi angulorum in analysi Diophantea ostenditur
523On three square numbers, of which the sum and the sum of products two apiece will be a square
541The expansion of the infinite product (1-x)(1-xx)(1-x3)(1-x4)(1-x5) etc. into a single series
542On the remarkable properties of the pentagonal numbers
552Observations about the division of squares by prime numbers
554A more exact disquisition about the residues remaining from the division of squares and of higher powers by prime numbers
556On the criteria of whether equation fxx + gyy = hxx admits a resolution or not
557De quibusdam eximiis proprietatibus circa divisores potestatum occurrentibus
558Proposita quacunque protressione ab unitate incipiente, quaeritur quot eius terminos a dminimum addi oporteat, ut omnes numeri producantur
559New assistance for solving the formula axx + 1 = yy
560Miscellaneous analyses
564Speculations about certain outstanding properties of numbers
566De inductione ad plenam certitudinem evehenda
586Considerations about a theorem of Fermat on the resolution of numbers into polygonal numbers
591On the relation between three and more quantities which are to be instituted
596On the sum of the series of numbers of the form 1/3 - 1/5 + 1/7 + 1/11 - 1/13 ... in which the prime numbers of the form 4n-1 have positive signs, and those of the form 4n+1 have negative signs
598De insigni promotione scientiae numerorum
610New demonstrations about the divisors of numbers of the form xx + nyy
683On a singular type of Diophantine questions and a most recondite method by which they are to be resolved
696On the cases in which the form x4 + kxxyy + y4 is permitted to be reduced to a square
699Inquiring on whether or not the number 100009 is prime
702De novo genere quaestionum arithmeticarum pro quibus solvendis certa methodus adhuc desideratur
708On forms of the type mxx + nyy for exploring prime numbers by idoneals of them with remarkable properties
713An investigation of a triangle in which the distances of the angles from the center of gravity of it may be expressed rationally
715On various ways of examining very large numbers, for whether or not they are primes
716The resolution of the Diophantine formula ab(maa+nbb) = cd(mcc+ndd) by rational numbers
718An easy method of finding several rather large prime numbers
719A more general method by which all adequately large numbers may be scrutinized for whether or not they are prime
725An illustration of a paradox about the idoneal, or suitable, numbers
732An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally
739An easy rule for Diophantine problems which are to be resolved quickly by integral numbers
744On divisors of numbers of the form mxx + nyy
748Investigatio quadrilateri, in quo singularum angulorum sinus datam inter se teneant rationem, ubi artificia prorsus singularia in Analysi Diophantea occurrunt
753Solution succincta et elegans problematis, quo quaeruntur tres numeri tales, ut tam summae quam differentiae binorum sint quadrata
754On a problem of geometry resolved by Diophantine analysis
755On cases for which the formula x4 + mxxyy + y4 can be reduced to a square
758De binis formulis speciei xx + myy et xx + nyy inter se concordibus et discordibus
763On finding three or more numbers, the sum of which is a square, while the sum of the squares is a fourth power
764An easy resolution to a most difficult question, where this most general form vvzz(axx+byy)2 + Δxxyy(avv+bzz)2 is required to be reduced to a square
769A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange
772On a notable advancement of Diophantine analysis
773A solution of a most difficult problem, in which the two forms aaxx + bbyy et aayy + bbxx must be rendered into squares
774An investigation of two numbers of the form xy(x4-y4), of which the product and the quotient will be a square
775On two numbers, of which the sum when increased or decreased by the square of one of them produces a square
776Elucidations about two sums of pairs of biquadratics, which are mutually equal
777On the resolution of the equation 0 = a + bx + cy + dxx + exy + fyy + gxxy + hyy + ixxyy by rational numbers
778A new and easy method for reducing cubic and biquadratic forms to squares
792Tractatus de numerorum doctrina capita sedecim, quae supersunt
793Thoughts concerning Diophantine analysis
796Research into the problem of three square numbers such that the sum of any two less the third one provides a square number
797Further and curious research into the problem of four positive numbers and an arithmetical proportion such that the sum of any two is always a square number
798On amicable numbers
799A fragment of a commentary, the most part on finding the relation between the sides of triangles of which the area is able to be expressed rationally, and of triangles in which the lines from each angle bisecting the opposite line are rationals