“Congruent number”的意思、由来-开放百科全书

In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.^[1] A more general definition includes all positive rational numbers with this property.^[2]

For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 is not a congruent number.

If {{mvar|q}} is a congruent number then {{math|s²q}} is also a congruent number for any natural number {{mvar|s}} (just by multiplying each side of the triangle by {{mvar|s}}), and vice versa. This leads to the observation that whether a nonzero rational number {{mvar|q}} is a congruent number depends only on its residue in the group

Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers, when speaking about congruent numbers.

Congruent number problem

The question of determining whether a given rational number is a congruent number is called the congruent number problem. This problem has not (as of 2016) been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Relation to elliptic curves

The question of whether a given number is congruent turns out to be equivalent to the condition that a certain elliptic curve has positive rank.^[2] An alternative approach to the idea is presented below (as can essentially also be found in the introduction to Tunnell's paper).

Suppose {{mvar|a}}, {{mvar|b}}, {{mvar|c}} are numbers (not necessarily positive or rational) which satisfy the following two equations:

and {{mvar|y}} is not 0 (if {{math|y {{=}} 0}} then {{math|a {{=}} -c}}, so {{math|b {{=}} 0}}, but {{math|({{frac|1|2}})ab {{=}} n}} is nonzero, a contradiction).

Conversely, if {{mvar|x}} and {{mvar|y}} are numbers which satisfy the above equation and {{mvar|y}} is not 0, set

These two correspondences between ({{mvar|a}},{{mvar|b}},{{mvar|c}}) and ({{mvar|x}},{{mvar|y}}) are inverses of each other, so

we have a one-to-one correspondence between any solution of the two equations in

from the formulas in the two correspondences, for rational {{mvar|n}} we see that {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} are

rational if and only if the corresponding {{mvar|x}} and {{mvar|y}} are rational, and vice versa.

(We also have that {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} are all positive if and only if {{mvar|x}} and {{mvar|y}} are all positive;

that if {{mvar|x}} and {{mvar|y}} are positive then {{math|x² - n²}} must be positive, so the formula for

Thus a positive rational number {{mvar|n}} is congruent if and only if the equation

It can be shown (as a nice application of Dirichlet's theorem on primes in arithmetic progression)

that the only torsion points on this elliptic curve are those with {{mvar|y}} equal to 0, hence the

existence of a rational point with {{mvar|y}} nonzero is equivalent to saying the elliptic curve has positive rank.

Another approach to solving is to start with integer value of n denoted as N and solve

Current progress

For example, it is known^[6] that for a prime number {{mvar|p}}, the following holds:

It is also known^[7] that in each of the congruence classes {{math|5, 6, 7 (mod 8)}}, for any given {{mvar|k}} there are infinitely many square-free congruent numbers with {{mvar|k}} prime factors.

Notes

1. ^{{MathWorld |urlname=CongruentNumber |title=Congruent Number}}
2. ^¹{{citation |first = Neal |last = Koblitz |authorlink = Neal Koblitz |title = Introduction to Elliptic Curves and Modular Forms |isbn = 0-387-97966-2 |publisher = Springer-Verlag |year = 1993| page = 3 |location = New York}}
3. ^{{citation|title=Number Theory and Its History|first=Øystein|last=Ore|authorlink=Øystein Ore|publisher=Courier Dover Corporation|year=2012|isbn=978-0-486-13643-1|pages=202–203|url=https://books.google.com/books?id=beC7AQAAQBAJ&pg=PA202}}.
4. ^{{citation|first=Keith|last=Conrad|title=The congruent number problem|journal=Harvard College Mathematical Review|url=http://www.thehcmr.org/issue2_2/congruent_number.pdf|volume=2|issue=2|date=Fall 2008|pages=58–73|deadurl=yes|archiveurl=https://web.archive.org/web/20130120090003/http://www.thehcmr.org/issue2_2/congruent_number.pdf|archivedate=2013-01-20|df=}}.
5. ^{{citation|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|first=David|last=Darling|publisher=John Wiley & Sons|year=2004|isbn=978-0-471-66700-1|page=77|url=https://books.google.com/books?id=HrOxRdtYYaMC&pg=PA77}}.
6. ^{{citation |author=Paul Monsky |title=Mock Heegner Points and Congruent Numbers |journal=Mathematische Zeitschrift |volume=204 |issue=1 |year=1990 |pages=45–67 |doi=10.1007/BF02570859}}
7. ^{{citation | last = Tian | first = Ye | authorlink = Tian Ye (mathematician) | arxiv = 1210.8231 | doi = 10.4310/CJM.2014.v2.n1.a4 | issue = 1 | journal = Cambridge Journal of Mathematics | mr = 3272014 | pages = 117–161 | title = Congruent numbers and Heegner points | volume = 2 | year = 2014}}.

Congruent number problem

Relation to elliptic curves

Current progress

Notes

References

External links