1、欧几里得的黄金时代(300BCE)到丢番图的白银时代(250BCE-250CE)。----丢番图被称为【代数之父】最著名的就是他的《算术》一书讲了有关代数方程求解问题和数的理论。----丢番图的工作是在古希腊亚历山大城完成的。那个时候亚历山大城是世界数学的中心。----丢番图所在的大时代250BCE-350CE也被称为数学的白银时代或者后亚历山大城时代。这个时代很多数学发现被一直用到当代。丢番图本人大概在250CE左右----欧几里得所在的300BE年代则属于数学的黄金时代。欧几里得比丢番图早了年长500-600年。----欧几里得把几何学搞得登峰造极而丢番图则是代数的开创者。2、丢番图使用符号(希腊字母)来表示未知数使用符号作为乘法的算术运算符号。----丢番图还事实上发明了数论。Arithmetica is also significant for its results in the theory of numbers, such as the fact that no integer of the form 8n7 an be written as the sum of three squares.3、丢番图的代数研究工作后继有人----许多都是1000多年之后。----The solution in integers of algebraic equations in more than one unkown with integral coefficients is one of the most difficult problem in the theory of numbers. 具有积分系数的多个未知数代数方程的整数解是数论中最困难的问题之一。----The study of diophantine equations involves an interplay among number theory, calculus, combinatorics, algebra and geometry. 丢番图方程的研究涉及数论、微积分、组合学、代数和几何之间的相互作用。4、丢番图方程一共有200个左右。比如这一文章说的是189个。----丢番图有三本著作其中最有名的是《算术/Arithmetica》当中包含了189个问题及答案而许多都是不定方程组(变量的个数大于方程的个数)或者不定方程式(两个变量以上但只有一个方程)。这种不定方程要解决三个问题1是判断何时有解。2是有解的时候决定解的个数。3是求出所有解。----鸡兔同笼问题今有雉兔同笼上有三十五头下有九十四足问雉兔各几何----百鸡问题鸡翁一直钱五鸡母一直钱三鸡雏三直钱一。百钱买百鸡问鸡翁、母、雏各几何----像这种鸡兔同笼和百鸡问题大概都是定解在丢番图的问题里面也是有的。只不过如上面的介绍这种问题在它的问题列表中属于简单的低一档的。6、丢番图方程这里说的是超过200个。----丢番图的著作算术中列出了超过200个代数问题以及其求解大概在公元250左右开始。----和上面一样也说的是使用希腊字母来表示未知数。In it he introduced algebraic manipulations on equations including a symbol for one unknown.----丢番图的著作还使用希腊字母表示未知数的一些幂操作(二次方、三次方、四次方、五次方、等于符号减号等。这已经是希腊数学的传统了。Like all Greeks at the time, Diophantus used the extended Greek alphabet to denote numbers.7、丢番图方程的具体例子----丢番图算术的第一卷给出了43个左右的方程式看其中前面几个是如何求解的。8、丢番图方程的求解实例--前三个。9、丢番图的方程全集(卷1共43个方程)----Problem 1 To split a given number (100) in two parts having a given difference (40).----Problem 2 To split a given number (60) in two parts having a given ratio (3:1).----Problem 3 To split a given number (80) in two parts, the larger of which exceeds a given ratio (3:1) of the smaller by a given number (4).----Problem 4 To find two numbers in a given ratio (5:1) and having a given difference (20).----Problem 5 To split a given number (100) in two parts such that given fractions of them (1/3, 1/5) have a given sum (30).----Problem 6 To split a given number (100) in two parts such that given fractions of them (1/4, 1/6) have a given difference (20).----Problem 7 To find a number such that when two given numbers (100,20) are subtracted from it, the remaining parts have a given ratio (1:3).----Problem 8 To find a number such that when two given numbers (100,20) are added to it, the sums have a given ratio (3:1).----Problem 9 To find a number which when subtracted from two given numbers (100,20), the remaining parts have a given ratio (6:1).----Problem 10 To find a number such that when added to a given number (20) and subtracted from another (100), the results have a given ratio (1:4).----Problem 11 To find a number such that when a given number (20) is added to it, and another (100) subtracted from it, the results have a given ratio (3:1).----Problem 12 To split a given number (100) twice into two pairs such that the first of each pair have a given ratio (2:1) as do the seconds (3:1).----Problem 13 To split a given number (100) as three pairs such that the second number of each pair and the first of the next pair (in cyclic order) have given ratios (3:1, 2:1, 4:1).----Problem 14 To find two numbers whose product and sum have a given ratio (3:1).----Problem 15 To find two numbers such that when a given number (30) istransferred from the second to the first, and when another number (50) is transferred from the first to the second, the resulting pairs have given ratios (2:1,3:1).----Problem 16 To find three numbers such that the sum of any two are given (20,30,40).----Problem 17 To find four numbers such that the sums of any three of them are given (20,22,24,27).----Problem 18 To find three numbers such that the sum of any two is greater than the third by given amounts (20,30,40).----Problem 19 is the same as 18, solved with a different method.----Problem 20 To find four numbers such that the sum of any three is greater than the fourth by given amounts (20,30,40,50).----Problem 21 is the same as 20, solved with a different method.----Problem 22 To split a given number (100) in three parts such the first two together have a given ratio to the third (3:1) and the last two together have a given ratio to the first (4:1)----Problem 23 To find three numbers such that the third exceeds the second by a given fraction (1/3) of the first; the second exceeds the first by a fraction (1/3) of the third; and the first exceeds a fraction (1/3) of the second by a given amount (10).----Problem 24 is the same as 23, solved with a different method.----Problem 25 To find three numbers such that if each transfers a given fraction of itself (1/3,1/4,1/5) to the next (cyclically), the results are all equal.----Problem 26 To find four numbers such that if each transfers a given fraction of itself (1/3,1/4,1/5,1/6) to the next (cyclically), the results are all equal.----Problem 27 To find three numbers such that if each receives a given fraction of the sum of the remaining numbers (1/3,1/4,1/5), the results are all equal.----Problem 28 To find four numbers such that if each receives a given fraction of the sum of the remaining numbers (1/3,1/4,1/5,1/6), the results are all equal.----Problem 29 To find a number which when multiplied by two given numbers (200,5) makes one a square and the other its side.----Problem 30 To find two numbers whose sum and product are given (20,96).----Problem 31 To find two numbers whose sum and sum of squares are given (20,208).----Problem 32 To find two numbers whose sum and difference of squares are given (20,80).----Problem 33 To find two numbers whose difference and product are given (4,96).----Problem 34 To find two numbers in a given ratio (3:1) such that their sum and the sum of their squares have a given ratio (1:5).----Problem 35 To find two numbers in a given ratio (3:1) such that their difference and the sum of their squares have a given ratio (1:10).----Problem 36 To find two numbers in a given ratio (3:1) such that their sum and the difference of their squares have a given ratio (1:6).----Problem 37 To find two numbers in a given ratio (3:1) such that their difference and the difference of their squares have a given ratio (1:12).----Problem 38 To find two numbers in a given ratio (3:1) such that the smaller square and the larger number have a given ratio (6:1).----Problem 39 To find two numbers in a given ratio (3:1) such that the smaller square and its side have a given ratio (6:1).----Problem 40 To find two numbers in a given ratio (3:1) such that the smaller square and their sum have a given ratio (2:1).----Problem 41 To find two numbers in a given ratio (3:1) such that the smaller square and their difference have a given ratio (6:1).----Problem 42 restates problems 38 to 41 for the larger square.----Problem 43 To find a number, together with two given numbers (3,5), such that when any two are added together then multiplied with the third, the result is three numbers which have equal differences.9、丢番图的方程全集(卷6共24个方程)----Problem 1 To find a right-angled triangle such that each side subtracted from the hypotenuse is a cube.----Problem 2 To find a right-angled triangle such that each side added to the hypotenuse is a cube.----Problem 3 To find a right-angled triangle whose area added to a given number (5) is a square.----Problem 4 To find a right-angled triangle whose area minus a given number (6) is a square.----Problem 5 To find a right-angled triangle such that its area subtracted from a given number (10) gives a square.----Problem 6 To find a right-angled triangle such that a side added to its area is a given number (7).----Problem 7 To find a right-angled triangle such that its side subtracted from its area is a given number (7).----Problem 8 To find a right-angled triangle such that the sum of the sides added to the area is a given number (6).----Problem 9 To find a right-angled triangle such that the sum of the sides subtracted from the area is a given number (6).----Problem 10 To find a right-angled triangle such that the sum of a side with the hypotenuse and the area is a given number (4).----Problem 11 To find a right-angled triangle such that the sum of a side with the hypotenuse subtracted from the area is a given number (4).----Problem 12 To find a right-angled triangle such that the area added to any side is a square.----Problem 13 To find a right-angled triangle such that the area minus any side is a square.----Problem 14 To find a right-angled triangle such that the area minus the hypotenuse or minus a side are each squares.----Problem 15 To find a right-angled triangle such that the area added to the hypotenuse or a side are each squares.----Problem 16 To find a right-angled triangle such that the bisector of an acute angle cuts the opposite side rationally.----Problem 17 To find a right-angled triangle such that the area added to the hypotenuse is a square and the perimeter is a cube.----Problem 18 To find a right-angled triangle such that the area added to the hypotenuse is a cube and the perimeter is a square.----Problem 19 To find a right-angled triangle such that its area added to a side is a square and the perimeter is a cube.----Problem 20 To find a right-angled triangle such that its area added to a side is a cube and the perimeter is a square.----Problem 21 To find a right-angled triangle such that its area added to the perimeter is a cube and the perimeter is a square.----Problem 22 To find a right-angled triangle such that its area added to the perimeter is a square and the perimeter is a cube.----Problem 23 To find a right-angled triangle such that the square of the hypotenuse is the sum of a square and its side, while the square of the hypotenuse divided by a side is the sum of a cube and its side.----Problem 24 To find a right-angled triangle such that one side is a cube, the other side is the difference between a cube and its side, and the hypotenuse is the sum of a cube and its side.10、丢番图算术第六卷的部分解----Problem 1 To find a right-angled triangle such that each side subtracted from the hypotenuse is a cube.----Problem 2 To find a right-angled triangle such that each side added to the hypotenuse is a cube.----Problem 3 To find a right-angled triangle whose area added to a given number (5) is a square.----Problem 23 To find a right-angled triangle such that the square of the hypotenuse is the sum of a square and its side, while the square of the hypotenuse divided by a side is the sum of a cube and its side.----Problem 24 To find a right-angled triangle such that one side is a cube, the other side is the difference between a cube and its side, and the hypotenuse is the sum of a cube and its side.完摘要古希腊数学家丢番图约250CE被誉为代数之父其代表作《算术》开创了代数方程求解和数论研究。作为亚历山大数学白银时代的代表人物他比黄金时代的欧几里得晚约500年。该书包含189个代数问题涉及不定方程求解包括著名的鸡兔同笼类问题。丢番图首次使用希腊字母表示未知数和运算符号建立了完整的代数符号体系。其方程研究涉及数论、几何等多领域交叉提出的丢番图方程问题至今仍是数论难题。