4047:
3826:
3081:
580:
850:. A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is
3092:
states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain distance during an arbitrary time interval, it will cover 3, 5, 7, etc.
3985:
809:
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is
2557:
The difference of two squares can also be used as an arithmetical short cut. If two numbers (whose average is a number which is easily squared) are multiplied, the difference of two squares can be used to give you the product of the original two numbers.
3067:
2909:
2187:
4200:
3884:
2280:
2362:
1620:
1213:
3417:
1454:
3569:
2488:
2426:
245:
2101:
2057:
3151:
2621:
2920:
1020:
804:
372:
119:
2771:
3681:
3366:
3632:
469:
2739:
1849:
1268:
3303:
2521:
1906:
1759:
684:
4071:
3803:
3762:
2693:
2650:
848:
628:
513:
1315:
947:
731:
1795:
1712:
1676:
1075:
554:
293:
1979:
1944:
3471:
3444:
3270:
2914:
Therefore, the difference of two consecutive perfect squares is an odd number. Similarly, the difference of two arbitrary perfect squares is calculated as follows:
3180:
1025:
3313:
Several algorithms in number theory and cryptography use differences of squares to find factors of integers and detect composite numbers. A simple example is the
900:
874:
3223:
2541:
3721:
3701:
3652:
3589:
3243:
3200:
2109:
1988:, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.
3980:{\displaystyle {\mathbf {a} }\cdot {\mathbf {a} }-{\mathbf {b} }\cdot {\mathbf {b} }=({\mathbf {a} }+{\mathbf {b} })\cdot ({\mathbf {a} }-{\mathbf {b} })}
2195:
4018:. This follows from the left side of the equation being equal to zero, requiring the right side to equal zero as well, and so the vector sum of
4364:
3072:
Therefore, the difference of two even perfect squares is a multiple of 4 and the difference of two odd perfect squares is a multiple of 8.
2288:
1465:
4343:
1083:
4380:
3371:
42:(multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the
1323:
3814:
3099:
949:. Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore,
3089:
4396:
3476:
1997:
4401:
4237:
2434:
2370:
377:
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the
4242:
3314:
4046:
159:
2065:
2021:
1024:
3062:{\displaystyle {\begin{array}{lcl}(n+k)^{2}-n^{2}&=&((n+k)+n)((n+k)-n)\\&=&k(2n+k)\end{array}}}
2552:
1044:
that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial
586:
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a
2567:
2904:{\displaystyle {\begin{array}{lcl}(n+1)^{2}-n^{2}&=&((n+1)+n)((n+1)-n)\\&=&2n+1\end{array}}}
952:
736:
304:
51:
4315:
43:
3305:), thus the distance from the starting point are consecutive squares for integer values of time elapsed.
3810:
3657:
3320:
2755:
3597:
590:. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e.
416:
410:
is commutative. To see this, apply the distributive law to the right-hand side of the equation and get
378:
4038:(the short diagonal of the rhombus) must equal zero, which indicates the diagonals are perpendicular.
2698:
4406:
1802:
1221:
146:
3275:
4253:
4003:
3863:
3859:
3825:
2496:
1855:
1718:
633:
587:
3080:
3767:
3726:
2658:
2629:
813:
593:
478:
392:
142:
125:
4276:
2011:
from expressions (or at least moving them), applying to division by some combinations involving
1273:
905:
689:
4209:
Historically, the
Babylonians used the difference of two squares to calculate multiplications.
4360:
4354:
4339:
4333:
4007:
2001:
1985:
1764:
4301:
1684:
1648:
1047:
521:
260:
4227:
3875:
1949:
1914:
385:
150:
39:
3449:
3422:
3248:
579:
4232:
3806:
3156:
2182:{\displaystyle ={\dfrac {5}{{\sqrt {3}}+4}}\times {\dfrac {{\sqrt {3}}-4}{{\sqrt {3}}-4}}}
251:
879:
853:
3205:
2526:
4288:
3852:
3706:
3686:
3637:
3574:
3228:
3185:
1639:
1631:
630:. The area of the shaded part can be found by adding the areas of the two rectangles;
4390:
4258:
4195:{\displaystyle a^{n}-b^{n}=(a-b){\biggl (}\sum _{k=0}^{n-1}a^{n-1-k}b^{k}{\biggr )}.}
2925:
2776:
2751:
17:
3867:
4015:
3871:
2012:
2004:
31:
1041:
2275:{\displaystyle ={\dfrac {5({\sqrt {3}}-4)}{({\sqrt {3}}+4)({\sqrt {3}}-4)}}}
4247:
3096:
From the equation for uniform linear acceleration, the distance covered
3093:
times that distance in the subsequent time intervals of the same length.
2008:
3805:. This forms the basis of several factorization algorithms (such as the
1040:
The formula for the difference of two squares can be used for factoring
4011:
3202:(acceleration due to gravity without air resistance), and time elapsed
4028:(the long diagonal of the rhombus) dotted with the vector difference
1459:
Moreover, this formula can also be used for simplifying expressions:
145:
of the factorization identity is straightforward. Starting from the
2357:{\displaystyle ={\dfrac {5({\sqrt {3}}-4)}{{\sqrt {3}}^{2}-4^{2}}}}
1615:{\displaystyle (a+b)^{2}-(a-b)^{2}=(a+b+a-b)(a+b-a+b)=(2a)(2b)=4ab}
4250:, the shared difference of three squares in arithmetic progression
4045:
4006:(which means that their dot squares are equal), this demonstrates
3824:
3079:
4050:
Visual proof of the differences between two squares and two cubes
1208:{\displaystyle x^{4}-1=(x^{2}+1)(x^{2}-1)=(x^{2}+1)(x+1)(x-1)}
1036:
Factorization of polynomials and simplification of expressions
3412:{\displaystyle a_{i}:=\left\lceil {\sqrt {N}}\right\rceil +i}
2553:
Multiplication algorithm § Quarter square multiplication
578:
4303:
The
Mechanical Universe: Introduction to Mechanics and Heat
1449:{\displaystyle x^{2}-y^{2}+x-y=(x+y)(x-y)+x-y=(x-y)(x+y+1)}
3088:
A ramification of the difference of consecutive squares,
3119:
1996:
The difference of two squares can also be used in the
4074:
3887:
3770:
3729:
3709:
3689:
3660:
3640:
3600:
3577:
3479:
3452:
3425:
3374:
3323:
3278:
3251:
3231:
3208:
3188:
3159:
3102:
2923:
2774:
2701:
2661:
2632:
2570:
2529:
2499:
2445:
2437:
2378:
2373:
2296:
2291:
2203:
2198:
2144:
2117:
2112:
2070:
2068:
2026:
2024:
1952:
1917:
1858:
1805:
1767:
1721:
1687:
1651:
1468:
1326:
1276:
1224:
1086:
1050:
955:
908:
882:
856:
816:
739:
692:
636:
596:
524:
481:
419:
307:
263:
162:
54:
4359:(5th ed.). Cengage Learning. pp. 467–469.
3564:{\displaystyle N=a_{i}^{2}-b^{2}=(a_{i}+b)(a_{i}-b)}
4194:
3990:The proof is identical. For the special case that
3979:
3797:
3756:
3715:
3695:
3675:
3646:
3626:
3583:
3563:
3465:
3438:
3411:
3360:
3297:
3264:
3237:
3217:
3194:
3174:
3145:
3061:
2903:
2733:
2687:
2644:
2615:
2535:
2515:
2483:{\displaystyle =-{\dfrac {5({\sqrt {3}}-4)}{13}}.}
2482:
2421:{\displaystyle ={\dfrac {5({\sqrt {3}}-4)}{3-16}}}
2420:
2356:
2274:
2181:
2095:
2051:
1973:
1938:
1900:
1843:
1789:
1753:
1706:
1670:
1630:The difference of two squares is used to find the
1614:
1448:
1309:
1262:
1207:
1069:
1014:
941:
894:
868:
842:
798:
725:
678:
622:
548:
507:
463:
366:
287:
239:
113:
4184:
4118:
3771:
3730:
3571:is a (potentially non-trivial) factorization of
1984:Since the two factors found by this method are
1678:can be found using difference of two squares:
4353:Tussy, Alan S.; Gustafson, Roy David (2011).
3594:This trick can be generalized as follows. If
2746:Difference of two consecutive perfect squares
8:
1218:As a second example, the first two terms of
240:{\displaystyle (a+b)(a-b)=a^{2}+ba-ab-b^{2}}
4291:TheMathPage.com, retrieved 22 December 2011
4279:TheMathPage.com, retrieved 22 December 2011
2096:{\displaystyle {\dfrac {5}{{\sqrt {3}}+4}}}
2052:{\displaystyle {\dfrac {5}{{\sqrt {3}}+4}}}
3317:, which considers the sequence of numbers
3146:{\displaystyle s=ut+{\tfrac {1}{2}}at^{2}}
4183:
4182:
4176:
4154:
4138:
4127:
4117:
4116:
4092:
4079:
4073:
3968:
3967:
3958:
3957:
3942:
3941:
3932:
3931:
3919:
3918:
3909:
3908:
3899:
3898:
3889:
3888:
3886:
3769:
3728:
3708:
3688:
3659:
3639:
3618:
3605:
3599:
3576:
3546:
3524:
3508:
3495:
3490:
3478:
3457:
3451:
3430:
3424:
3392:
3379:
3373:
3346:
3341:
3328:
3322:
3289:
3277:
3256:
3250:
3230:
3207:
3187:
3158:
3137:
3118:
3101:
2957:
2944:
2924:
2922:
2808:
2795:
2775:
2773:
2719:
2706:
2700:
2679:
2666:
2660:
2631:
2569:
2528:
2500:
2498:
2454:
2444:
2436:
2387:
2377:
2372:
2344:
2331:
2324:
2305:
2295:
2290:
2252:
2233:
2212:
2202:
2197:
2162:
2147:
2143:
2123:
2116:
2111:
2076:
2069:
2067:
2032:
2025:
2023:
1951:
1916:
1857:
1835:
1813:
1804:
1772:
1766:
1745:
1729:
1720:
1692:
1686:
1656:
1650:
1510:
1485:
1467:
1344:
1331:
1325:
1275:
1242:
1229:
1223:
1160:
1135:
1113:
1091:
1085:
1055:
1049:
973:
960:
954:
907:
881:
855:
834:
821:
815:
757:
744:
738:
691:
635:
614:
601:
595:
523:
499:
486:
480:
455:
424:
418:
358:
345:
306:
262:
231:
200:
161:
72:
59:
53:
2616:{\displaystyle 27\times 33=(30-3)(30+3)}
391:Conversely, if this identity holds in a
4269:
4062:are two elements of a commutative ring
1626:Complex number case: sum of two squares
3723:is composite with non-trivial factors
1015:{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
799:{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
367:{\displaystyle (a+b)(a-b)=a^{2}-b^{2}}
114:{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}
2626:Using the difference of two squares,
7:
4338:. Infobase Publishing. p. 131.
27:Mathematical identity of polynomials
3676:{\displaystyle a\not \equiv \pm b}
3361:{\displaystyle x_{i}:=a_{i}^{2}-N}
3075:
2750:The difference of two consecutive
1911:Therefore, the linear factors are
1645:For example, the complex roots of
25:
4010:the fact that two diagonals of a
3627:{\displaystyle a^{2}\equiv b^{2}}
2765:+1. This can be seen as follows:
2493:Here, the irrational denominator
464:{\displaystyle a^{2}+ba-ab-b^{2}}
4218:64 Ă— 56 = 60² − 4² = 3584
4215:93 Ă— 87 = 90² − 3² = 8091
3969:
3959:
3943:
3933:
3920:
3910:
3900:
3890:
2734:{\displaystyle 30^{2}-3^{2}=891}
2059:can be rationalised as follows:
2018:For example: The denominator of
2007:. This is a method for removing
1023:
3809:) and can be combined with the
1844:{\displaystyle =z^{2}-(2i)^{2}}
1263:{\displaystyle x^{2}-y^{2}+x-y}
254:, the middle two terms cancel:
4332:Stanton, James Stuart (2005).
4113:
4101:
3974:
3954:
3948:
3928:
3792:
3774:
3751:
3733:
3558:
3539:
3536:
3517:
3298:{\displaystyle s\propto t^{2}}
3052:
3037:
3021:
3012:
3000:
2997:
2994:
2985:
2973:
2970:
2941:
2928:
2872:
2863:
2851:
2848:
2845:
2836:
2824:
2821:
2792:
2779:
2610:
2598:
2595:
2583:
2467:
2451:
2400:
2384:
2318:
2302:
2265:
2249:
2246:
2230:
2225:
2209:
1968:
1953:
1933:
1918:
1895:
1880:
1877:
1862:
1832:
1822:
1597:
1588:
1585:
1576:
1570:
1546:
1543:
1519:
1507:
1494:
1482:
1469:
1443:
1425:
1422:
1410:
1392:
1380:
1377:
1365:
1304:
1292:
1289:
1277:
1202:
1190:
1187:
1175:
1172:
1153:
1147:
1128:
1125:
1106:
1009:
997:
994:
982:
936:
924:
921:
909:
793:
781:
778:
766:
720:
708:
705:
693:
673:
661:
652:
640:
335:
323:
320:
308:
190:
178:
175:
163:
108:
96:
93:
81:
1:
3225:it follows that the distance
2516:{\displaystyle {\sqrt {3}}+4}
1901:{\displaystyle =(z+2i)(z-2i)}
1754:{\displaystyle =z^{2}-4i^{2}}
686:, which can be factorized to
679:{\displaystyle a(a-b)+b(a-b)}
4277:Complex or imaginary numbers
4042:Difference of two nth powers
3851: (blue) are shown with
3090:Galileo's law of odd numbers
3084:Galileo's law of odd numbers
3076:Galileo's law of odd numbers
1077:can be factored as follows:
4335:Encyclopedia of Mathematics
4243:Aurifeuillean factorization
3858:The identity also holds in
3815:Miller–Rabin primality test
3798:{\displaystyle \gcd(a+b,N)}
3757:{\displaystyle \gcd(a-b,N)}
3315:Fermat factorization method
2688:{\displaystyle a^{2}-b^{2}}
2645:{\displaystyle 27\times 33}
902:. This rectangle's area is
843:{\displaystyle a^{2}-b^{2}}
623:{\displaystyle a^{2}-b^{2}}
508:{\displaystyle a^{2}-b^{2}}
4423:
2550:
1992:Rationalising denominators
1310:{\displaystyle (x+y)(x-y)}
942:{\displaystyle (a+b)(a-b)}
726:{\displaystyle (a+b)(a-b)}
398:for all pairs of elements
4381:difference of two squares
4238:Sophie Germain's identity
3309:Factorization of integers
2523:has been rationalised to
36:difference of two squares
4316:"Babylonian mathematics"
3446:equals a perfect square
1790:{\displaystyle i^{2}=-1}
475:For this to be equal to
1707:{\displaystyle z^{2}+4}
1671:{\displaystyle z^{2}+4}
1070:{\displaystyle x^{4}-1}
549:{\displaystyle ba-ab=0}
384:The proof holds in any
288:{\displaystyle ba-ab=0}
4196:
4149:
4051:
3981:
3855:
3799:
3758:
3717:
3697:
3677:
3648:
3628:
3585:
3565:
3467:
3440:
3413:
3362:
3299:
3266:
3239:
3219:
3196:
3182:constant acceleration
3176:
3147:
3085:
3063:
2905:
2754:is the sum of the two
2735:
2689:
2646:
2617:
2537:
2517:
2484:
2422:
2358:
2276:
2183:
2097:
2053:
1975:
1974:{\displaystyle (z-2i)}
1940:
1939:{\displaystyle (z+2i)}
1902:
1845:
1791:
1755:
1708:
1672:
1638:of two squares, using
1616:
1450:
1311:
1264:
1209:
1071:
1016:
943:
896:
870:
844:
800:
727:
680:
624:
583:
550:
509:
465:
368:
289:
241:
115:
4197:
4123:
4049:
3982:
3828:
3813:to give the stronger
3811:Fermat primality test
3800:
3759:
3718:
3698:
3678:
3649:
3629:
3586:
3566:
3468:
3466:{\displaystyle b^{2}}
3441:
3439:{\displaystyle x_{i}}
3414:
3363:
3300:
3267:
3265:{\displaystyle t^{2}}
3240:
3220:
3197:
3177:
3148:
3083:
3064:
2906:
2736:
2690:
2647:
2618:
2538:
2518:
2485:
2423:
2359:
2277:
2184:
2098:
2054:
1976:
1941:
1903:
1846:
1792:
1756:
1709:
1673:
1617:
1451:
1312:
1265:
1210:
1072:
1017:
944:
897:
871:
845:
801:
728:
681:
625:
582:
551:
510:
466:
369:
290:
242:
116:
18:Difference of squares
4397:Algebraic identities
4289:Multiplying Radicals
4072:
3885:
3860:inner product spaces
3768:
3727:
3707:
3687:
3658:
3638:
3598:
3575:
3477:
3450:
3423:
3372:
3321:
3276:
3249:
3229:
3206:
3186:
3175:{\displaystyle u=0,}
3157:
3100:
2921:
2772:
2699:
2659:
2630:
2568:
2527:
2497:
2435:
2371:
2289:
2196:
2110:
2066:
2022:
1950:
1915:
1856:
1803:
1765:
1719:
1685:
1649:
1466:
1324:
1274:
1222:
1084:
1048:
953:
906:
880:
876:and whose height is
854:
814:
737:
690:
634:
594:
522:
479:
417:
305:
261:
160:
52:
4402:Commutative algebra
4300:RP Olenick et al.,
4254:Conjugate (algebra)
3500:
3351:
3245:is proportional to
2652:can be restated as
1270:can be factored as
895:{\displaystyle a-b}
869:{\displaystyle a+b}
4356:Elementary Algebra
4192:
4052:
3977:
3856:
3795:
3754:
3713:
3693:
3673:
3644:
3624:
3581:
3561:
3486:
3463:
3436:
3409:
3358:
3337:
3295:
3262:
3235:
3218:{\displaystyle t,}
3215:
3192:
3172:
3153:for initial speed
3143:
3128:
3086:
3059:
3057:
2901:
2899:
2731:
2685:
2642:
2613:
2536:{\displaystyle 13}
2533:
2513:
2480:
2475:
2418:
2416:
2354:
2352:
2272:
2270:
2179:
2177:
2138:
2093:
2091:
2049:
2047:
1986:complex conjugates
1971:
1936:
1898:
1841:
1787:
1751:
1704:
1668:
1612:
1446:
1307:
1260:
1205:
1067:
1012:
939:
892:
866:
840:
796:
723:
676:
620:
584:
546:
505:
461:
381:in two variables.
364:
285:
237:
126:elementary algebra
111:
4366:978-1-111-56766-8
3876:Euclidean vectors
3841: (cyan) and
3716:{\displaystyle N}
3696:{\displaystyle N}
3647:{\displaystyle N}
3584:{\displaystyle N}
3397:
3238:{\displaystyle s}
3195:{\displaystyle a}
3127:
2547:Mental arithmetic
2505:
2474:
2459:
2415:
2392:
2351:
2329:
2310:
2269:
2257:
2238:
2217:
2176:
2167:
2152:
2137:
2128:
2090:
2081:
2046:
2037:
16:(Redirected from
4414:
4383:at mathpages.com
4370:
4349:
4320:
4319:
4312:
4306:
4298:
4292:
4286:
4280:
4274:
4228:Sum of two cubes
4201:
4199:
4198:
4193:
4188:
4187:
4181:
4180:
4171:
4170:
4148:
4137:
4122:
4121:
4097:
4096:
4084:
4083:
4037:
4027:
4001:
3995:
3986:
3984:
3983:
3978:
3973:
3972:
3963:
3962:
3947:
3946:
3937:
3936:
3924:
3923:
3914:
3913:
3904:
3903:
3894:
3893:
3850:
3840:
3835: (purple),
3834:
3804:
3802:
3801:
3796:
3763:
3761:
3760:
3755:
3722:
3720:
3719:
3714:
3702:
3700:
3699:
3694:
3682:
3680:
3679:
3674:
3653:
3651:
3650:
3645:
3633:
3631:
3630:
3625:
3623:
3622:
3610:
3609:
3590:
3588:
3587:
3582:
3570:
3568:
3567:
3562:
3551:
3550:
3529:
3528:
3513:
3512:
3499:
3494:
3472:
3470:
3469:
3464:
3462:
3461:
3445:
3443:
3442:
3437:
3435:
3434:
3419:. If one of the
3418:
3416:
3415:
3410:
3402:
3398:
3393:
3384:
3383:
3367:
3365:
3364:
3359:
3350:
3345:
3333:
3332:
3304:
3302:
3301:
3296:
3294:
3293:
3271:
3269:
3268:
3263:
3261:
3260:
3244:
3242:
3241:
3236:
3224:
3222:
3221:
3216:
3201:
3199:
3198:
3193:
3181:
3179:
3178:
3173:
3152:
3150:
3149:
3144:
3142:
3141:
3129:
3120:
3068:
3066:
3065:
3060:
3058:
3027:
2962:
2961:
2949:
2948:
2910:
2908:
2907:
2902:
2900:
2878:
2813:
2812:
2800:
2799:
2740:
2738:
2737:
2732:
2724:
2723:
2711:
2710:
2694:
2692:
2691:
2686:
2684:
2683:
2671:
2670:
2651:
2649:
2648:
2643:
2622:
2620:
2619:
2614:
2542:
2540:
2539:
2534:
2522:
2520:
2519:
2514:
2506:
2501:
2489:
2487:
2486:
2481:
2476:
2470:
2460:
2455:
2446:
2427:
2425:
2424:
2419:
2417:
2414:
2403:
2393:
2388:
2379:
2363:
2361:
2360:
2355:
2353:
2350:
2349:
2348:
2336:
2335:
2330:
2325:
2321:
2311:
2306:
2297:
2281:
2279:
2278:
2273:
2271:
2268:
2258:
2253:
2239:
2234:
2228:
2218:
2213:
2204:
2188:
2186:
2185:
2180:
2178:
2175:
2168:
2163:
2160:
2153:
2148:
2145:
2139:
2136:
2129:
2124:
2118:
2102:
2100:
2099:
2094:
2092:
2089:
2082:
2077:
2071:
2058:
2056:
2055:
2050:
2048:
2045:
2038:
2033:
2027:
1980:
1978:
1977:
1972:
1945:
1943:
1942:
1937:
1907:
1905:
1904:
1899:
1850:
1848:
1847:
1842:
1840:
1839:
1818:
1817:
1796:
1794:
1793:
1788:
1777:
1776:
1760:
1758:
1757:
1752:
1750:
1749:
1734:
1733:
1713:
1711:
1710:
1705:
1697:
1696:
1677:
1675:
1674:
1669:
1661:
1660:
1621:
1619:
1618:
1613:
1515:
1514:
1490:
1489:
1455:
1453:
1452:
1447:
1349:
1348:
1336:
1335:
1316:
1314:
1313:
1308:
1269:
1267:
1266:
1261:
1247:
1246:
1234:
1233:
1214:
1212:
1211:
1206:
1165:
1164:
1140:
1139:
1118:
1117:
1096:
1095:
1076:
1074:
1073:
1068:
1060:
1059:
1027:
1021:
1019:
1018:
1013:
978:
977:
965:
964:
948:
946:
945:
940:
901:
899:
898:
893:
875:
873:
872:
867:
849:
847:
846:
841:
839:
838:
826:
825:
805:
803:
802:
797:
762:
761:
749:
748:
732:
730:
729:
724:
685:
683:
682:
677:
629:
627:
626:
621:
619:
618:
606:
605:
571:is commutative.
555:
553:
552:
547:
514:
512:
511:
506:
504:
503:
491:
490:
470:
468:
467:
462:
460:
459:
429:
428:
386:commutative ring
379:AM–GM inequality
373:
371:
370:
365:
363:
362:
350:
349:
294:
292:
291:
286:
246:
244:
243:
238:
236:
235:
205:
204:
151:distributive law
120:
118:
117:
112:
77:
76:
64:
63:
21:
4422:
4421:
4417:
4416:
4415:
4413:
4412:
4411:
4387:
4386:
4377:
4367:
4352:
4346:
4331:
4328:
4323:
4314:
4313:
4309:
4299:
4295:
4287:
4283:
4275:
4271:
4267:
4233:Binomial number
4224:
4207:
4172:
4150:
4088:
4075:
4070:
4069:
4044:
4029:
4019:
3997:
3991:
3883:
3882:
3842:
3836:
3830:
3823:
3821:Generalizations
3807:quadratic sieve
3766:
3765:
3725:
3724:
3705:
3704:
3685:
3684:
3656:
3655:
3636:
3635:
3614:
3601:
3596:
3595:
3573:
3572:
3542:
3520:
3504:
3475:
3474:
3453:
3448:
3447:
3426:
3421:
3420:
3388:
3375:
3370:
3369:
3324:
3319:
3318:
3311:
3285:
3274:
3273:
3252:
3247:
3246:
3227:
3226:
3204:
3203:
3184:
3183:
3155:
3154:
3133:
3098:
3097:
3078:
3056:
3055:
3032:
3025:
3024:
2968:
2963:
2953:
2940:
2919:
2918:
2898:
2897:
2883:
2876:
2875:
2819:
2814:
2804:
2791:
2770:
2769:
2752:perfect squares
2748:
2715:
2702:
2697:
2696:
2675:
2662:
2657:
2656:
2628:
2627:
2566:
2565:
2555:
2549:
2525:
2524:
2495:
2494:
2447:
2433:
2432:
2404:
2380:
2369:
2368:
2340:
2323:
2322:
2298:
2287:
2286:
2229:
2205:
2194:
2193:
2161:
2146:
2122:
2108:
2107:
2075:
2064:
2063:
2031:
2020:
2019:
1994:
1948:
1947:
1913:
1912:
1854:
1853:
1831:
1809:
1801:
1800:
1768:
1763:
1762:
1741:
1725:
1717:
1716:
1688:
1683:
1682:
1652:
1647:
1646:
1628:
1506:
1481:
1464:
1463:
1340:
1327:
1322:
1321:
1272:
1271:
1238:
1225:
1220:
1219:
1156:
1131:
1109:
1087:
1082:
1081:
1051:
1046:
1045:
1038:
1033:
969:
956:
951:
950:
904:
903:
878:
877:
852:
851:
830:
817:
812:
811:
753:
740:
735:
734:
688:
687:
632:
631:
610:
597:
592:
591:
577:
575:Geometric proof
520:
519:
515:, we must have
495:
482:
477:
476:
451:
420:
415:
414:
354:
341:
303:
302:
259:
258:
252:commutative law
227:
196:
158:
157:
147:right-hand side
139:
137:Algebraic proof
134:
68:
55:
50:
49:
28:
23:
22:
15:
12:
11:
5:
4420:
4418:
4410:
4409:
4404:
4399:
4389:
4388:
4385:
4384:
4376:
4375:External links
4373:
4372:
4371:
4365:
4350:
4344:
4327:
4324:
4322:
4321:
4307:
4293:
4281:
4268:
4266:
4263:
4262:
4261:
4256:
4251:
4245:
4240:
4235:
4230:
4223:
4220:
4206:
4203:
4191:
4186:
4179:
4175:
4169:
4166:
4163:
4160:
4157:
4153:
4147:
4144:
4141:
4136:
4133:
4130:
4126:
4120:
4115:
4112:
4109:
4106:
4103:
4100:
4095:
4091:
4087:
4082:
4078:
4043:
4040:
3988:
3987:
3976:
3971:
3966:
3961:
3956:
3953:
3950:
3945:
3940:
3935:
3930:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3892:
3870:, such as for
3822:
3819:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3712:
3692:
3672:
3669:
3666:
3663:
3643:
3621:
3617:
3613:
3608:
3604:
3580:
3560:
3557:
3554:
3549:
3545:
3541:
3538:
3535:
3532:
3527:
3523:
3519:
3516:
3511:
3507:
3503:
3498:
3493:
3489:
3485:
3482:
3460:
3456:
3433:
3429:
3408:
3405:
3401:
3396:
3391:
3387:
3382:
3378:
3357:
3354:
3349:
3344:
3340:
3336:
3331:
3327:
3310:
3307:
3292:
3288:
3284:
3281:
3259:
3255:
3234:
3214:
3211:
3191:
3171:
3168:
3165:
3162:
3140:
3136:
3132:
3126:
3123:
3117:
3114:
3111:
3108:
3105:
3077:
3074:
3070:
3069:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3031:
3028:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2967:
2964:
2960:
2956:
2952:
2947:
2943:
2939:
2936:
2933:
2930:
2927:
2926:
2912:
2911:
2896:
2893:
2890:
2887:
2884:
2882:
2879:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2818:
2815:
2811:
2807:
2803:
2798:
2794:
2790:
2787:
2784:
2781:
2778:
2777:
2747:
2744:
2743:
2742:
2730:
2727:
2722:
2718:
2714:
2709:
2705:
2682:
2678:
2674:
2669:
2665:
2641:
2638:
2635:
2624:
2623:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2582:
2579:
2576:
2573:
2561:For example:
2551:Main article:
2548:
2545:
2532:
2512:
2509:
2504:
2491:
2490:
2479:
2473:
2469:
2466:
2463:
2458:
2453:
2450:
2443:
2440:
2429:
2428:
2413:
2410:
2407:
2402:
2399:
2396:
2391:
2386:
2383:
2376:
2365:
2364:
2347:
2343:
2339:
2334:
2328:
2320:
2317:
2314:
2309:
2304:
2301:
2294:
2283:
2282:
2267:
2264:
2261:
2256:
2251:
2248:
2245:
2242:
2237:
2232:
2227:
2224:
2221:
2216:
2211:
2208:
2201:
2190:
2189:
2174:
2171:
2166:
2159:
2156:
2151:
2142:
2135:
2132:
2127:
2121:
2115:
2104:
2103:
2088:
2085:
2080:
2074:
2044:
2041:
2036:
2030:
1993:
1990:
1970:
1967:
1964:
1961:
1958:
1955:
1935:
1932:
1929:
1926:
1923:
1920:
1909:
1908:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1851:
1838:
1834:
1830:
1827:
1824:
1821:
1816:
1812:
1808:
1798:
1786:
1783:
1780:
1775:
1771:
1748:
1744:
1740:
1737:
1732:
1728:
1724:
1714:
1703:
1700:
1695:
1691:
1667:
1664:
1659:
1655:
1642:coefficients.
1640:complex number
1632:linear factors
1627:
1624:
1623:
1622:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1513:
1509:
1505:
1502:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1474:
1471:
1457:
1456:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1347:
1343:
1339:
1334:
1330:
1317:, so we have:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1259:
1256:
1253:
1250:
1245:
1241:
1237:
1232:
1228:
1216:
1215:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1163:
1159:
1155:
1152:
1149:
1146:
1143:
1138:
1134:
1130:
1127:
1124:
1121:
1116:
1112:
1108:
1105:
1102:
1099:
1094:
1090:
1066:
1063:
1058:
1054:
1037:
1034:
1032:
1029:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
976:
972:
968:
963:
959:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
891:
888:
885:
865:
862:
859:
837:
833:
829:
824:
820:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
760:
756:
752:
747:
743:
733:. Therefore,
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
645:
642:
639:
617:
613:
609:
604:
600:
576:
573:
559:for all pairs
557:
556:
545:
542:
539:
536:
533:
530:
527:
502:
498:
494:
489:
485:
473:
472:
458:
454:
450:
447:
444:
441:
438:
435:
432:
427:
423:
375:
374:
361:
357:
353:
348:
344:
340:
337:
334:
331:
328:
325:
322:
319:
316:
313:
310:
296:
295:
284:
281:
278:
275:
272:
269:
266:
248:
247:
234:
230:
226:
223:
220:
217:
214:
211:
208:
203:
199:
195:
192:
189:
186:
183:
180:
177:
174:
171:
168:
165:
138:
135:
133:
130:
122:
121:
110:
107:
104:
101:
98:
95:
92:
89:
86:
83:
80:
75:
71:
67:
62:
58:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4419:
4408:
4405:
4403:
4400:
4398:
4395:
4394:
4392:
4382:
4379:
4378:
4374:
4368:
4362:
4358:
4357:
4351:
4347:
4345:0-8160-5124-0
4341:
4337:
4336:
4330:
4329:
4325:
4317:
4311:
4308:
4305:
4304:
4297:
4294:
4290:
4285:
4282:
4278:
4273:
4270:
4264:
4260:
4259:Factorization
4257:
4255:
4252:
4249:
4246:
4244:
4241:
4239:
4236:
4234:
4231:
4229:
4226:
4225:
4221:
4219:
4216:
4213:
4212:For example:
4210:
4204:
4202:
4189:
4177:
4173:
4167:
4164:
4161:
4158:
4155:
4151:
4145:
4142:
4139:
4134:
4131:
4128:
4124:
4110:
4107:
4104:
4098:
4093:
4089:
4085:
4080:
4076:
4067:
4065:
4061:
4057:
4048:
4041:
4039:
4036:
4032:
4026:
4022:
4017:
4016:perpendicular
4013:
4009:
4005:
4000:
3994:
3964:
3951:
3938:
3925:
3915:
3905:
3895:
3881:
3880:
3879:
3877:
3873:
3869:
3865:
3861:
3854:
3849:
3845:
3839:
3833:
3827:
3820:
3818:
3816:
3812:
3808:
3789:
3786:
3783:
3780:
3777:
3748:
3745:
3742:
3739:
3736:
3710:
3690:
3670:
3667:
3664:
3661:
3641:
3619:
3615:
3611:
3606:
3602:
3592:
3578:
3555:
3552:
3547:
3543:
3533:
3530:
3525:
3521:
3514:
3509:
3505:
3501:
3496:
3491:
3487:
3483:
3480:
3458:
3454:
3431:
3427:
3406:
3403:
3399:
3394:
3389:
3385:
3380:
3376:
3355:
3352:
3347:
3342:
3338:
3334:
3329:
3325:
3316:
3308:
3306:
3290:
3286:
3282:
3279:
3272:(in symbols,
3257:
3253:
3232:
3212:
3209:
3189:
3169:
3166:
3163:
3160:
3138:
3134:
3130:
3124:
3121:
3115:
3112:
3109:
3106:
3103:
3094:
3091:
3082:
3073:
3049:
3046:
3043:
3040:
3034:
3029:
3018:
3015:
3009:
3006:
3003:
2991:
2988:
2982:
2979:
2976:
2965:
2958:
2954:
2950:
2945:
2937:
2934:
2931:
2917:
2916:
2915:
2894:
2891:
2888:
2885:
2880:
2869:
2866:
2860:
2857:
2854:
2842:
2839:
2833:
2830:
2827:
2816:
2809:
2805:
2801:
2796:
2788:
2785:
2782:
2768:
2767:
2766:
2764:
2760:
2757:
2753:
2745:
2728:
2725:
2720:
2716:
2712:
2707:
2703:
2680:
2676:
2672:
2667:
2663:
2655:
2654:
2653:
2639:
2636:
2633:
2607:
2604:
2601:
2592:
2589:
2586:
2580:
2577:
2574:
2571:
2564:
2563:
2562:
2559:
2554:
2546:
2544:
2530:
2510:
2507:
2502:
2477:
2471:
2464:
2461:
2456:
2448:
2441:
2438:
2431:
2430:
2411:
2408:
2405:
2397:
2394:
2389:
2381:
2374:
2367:
2366:
2345:
2341:
2337:
2332:
2326:
2315:
2312:
2307:
2299:
2292:
2285:
2284:
2262:
2259:
2254:
2243:
2240:
2235:
2222:
2219:
2214:
2206:
2199:
2192:
2191:
2172:
2169:
2164:
2157:
2154:
2149:
2140:
2133:
2130:
2125:
2119:
2113:
2106:
2105:
2086:
2083:
2078:
2072:
2062:
2061:
2060:
2042:
2039:
2034:
2028:
2016:
2014:
2010:
2006:
2003:
1999:
1998:rationalising
1991:
1989:
1987:
1982:
1965:
1962:
1959:
1956:
1930:
1927:
1924:
1921:
1892:
1889:
1886:
1883:
1874:
1871:
1868:
1865:
1859:
1852:
1836:
1828:
1825:
1819:
1814:
1810:
1806:
1799:
1784:
1781:
1778:
1773:
1769:
1746:
1742:
1738:
1735:
1730:
1726:
1722:
1715:
1701:
1698:
1693:
1689:
1681:
1680:
1679:
1665:
1662:
1657:
1653:
1643:
1641:
1637:
1633:
1625:
1609:
1606:
1603:
1600:
1594:
1591:
1582:
1579:
1573:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1516:
1511:
1503:
1500:
1497:
1491:
1486:
1478:
1475:
1472:
1462:
1461:
1460:
1440:
1437:
1434:
1431:
1428:
1419:
1416:
1413:
1407:
1404:
1401:
1398:
1395:
1389:
1386:
1383:
1374:
1371:
1368:
1362:
1359:
1356:
1353:
1350:
1345:
1341:
1337:
1332:
1328:
1320:
1319:
1318:
1301:
1298:
1295:
1286:
1283:
1280:
1257:
1254:
1251:
1248:
1243:
1239:
1235:
1230:
1226:
1199:
1196:
1193:
1184:
1181:
1178:
1169:
1166:
1161:
1157:
1150:
1144:
1141:
1136:
1132:
1122:
1119:
1114:
1110:
1103:
1100:
1097:
1092:
1088:
1080:
1079:
1078:
1064:
1061:
1056:
1052:
1043:
1035:
1030:
1028:
1026:
1006:
1003:
1000:
991:
988:
985:
979:
974:
970:
966:
961:
957:
933:
930:
927:
918:
915:
912:
889:
886:
883:
863:
860:
857:
835:
831:
827:
822:
818:
807:
790:
787:
784:
775:
772:
769:
763:
758:
754:
750:
745:
741:
717:
714:
711:
702:
699:
696:
670:
667:
664:
658:
655:
649:
646:
643:
637:
615:
611:
607:
602:
598:
589:
581:
574:
572:
570:
566:
562:
543:
540:
537:
534:
531:
528:
525:
518:
517:
516:
500:
496:
492:
487:
483:
456:
452:
448:
445:
442:
439:
436:
433:
430:
425:
421:
413:
412:
411:
409:
405:
401:
397:
394:
389:
387:
382:
380:
359:
355:
351:
346:
342:
338:
332:
329:
326:
317:
314:
311:
301:
300:
299:
282:
279:
276:
273:
270:
267:
264:
257:
256:
255:
253:
232:
228:
224:
221:
218:
215:
212:
209:
206:
201:
197:
193:
187:
184:
181:
172:
169:
166:
156:
155:
154:
152:
148:
144:
136:
131:
129:
127:
105:
102:
99:
90:
87:
84:
78:
73:
69:
65:
60:
56:
48:
47:
46:
45:
41:
37:
33:
19:
4355:
4334:
4310:
4302:
4296:
4284:
4272:
4217:
4214:
4211:
4208:
4068:
4063:
4059:
4055:
4053:
4034:
4030:
4024:
4020:
4008:analytically
3998:
3992:
3989:
3868:real numbers
3857:
3847:
3843:
3837:
3831:
3593:
3312:
3095:
3087:
3071:
2913:
2762:
2758:
2749:
2625:
2560:
2556:
2492:
2017:
2013:square roots
2005:denominators
1995:
1983:
1910:
1644:
1635:
1629:
1458:
1217:
1039:
808:
585:
568:
564:
560:
558:
474:
407:
403:
399:
395:
390:
383:
376:
297:
249:
149:, apply the
140:
123:
35:
29:
4407:Subtraction
4002:have equal
3872:dot product
1042:polynomials
32:mathematics
4391:Categories
4326:References
2002:irrational
4165:−
4159:−
4143:−
4125:∑
4108:−
4086:−
3965:−
3952:⋅
3916:⋅
3906:−
3896:⋅
3862:over the
3740:−
3668:±
3612:≡
3553:−
3502:−
3353:−
3283:∝
3016:−
2951:−
2867:−
2802:−
2713:−
2695:which is
2673:−
2637:×
2590:−
2575:×
2462:−
2442:−
2409:−
2395:−
2338:−
2313:−
2260:−
2220:−
2170:−
2155:−
2141:×
1960:−
1887:−
1820:−
1782:−
1736:−
1559:−
1538:−
1501:−
1492:−
1417:−
1402:−
1387:−
1357:−
1338:−
1299:−
1255:−
1236:−
1197:−
1142:−
1098:−
1062:−
1004:−
967:−
931:−
887:−
828:−
788:−
751:−
715:−
668:−
647:−
608:−
532:−
493:−
449:−
440:−
352:−
330:−
271:−
225:−
216:−
185:−
103:−
66:−
4248:Congruum
4222:See also
3829:Vectors
3665:≢
3400:⌉
3390:⌈
298:leaving
153:to get
44:identity
4205:History
4066:, then
4012:rhombus
3703:, then
3473:, then
1761:(since
1634:of the
406:, then
250:By the
40:squared
4363:
4342:
3853:arrows
3368:, for
34:, the
4265:Notes
4004:norms
3864:field
2756:bases
2009:surds
1031:Usage
588:plane
567:, so
143:proof
132:Proof
38:is a
4361:ISBN
4340:ISBN
4058:and
4014:are
3996:and
3764:and
3683:mod
3654:and
3634:mod
2761:and
1946:and
402:and
393:ring
141:The
4054:If
3874:of
3866:of
3772:gcd
3731:gcd
2729:891
2000:of
1636:sum
124:in
30:In
4393::
4033:-
4023:+
3878::
3846:+
3817:.
3591:.
3386::=
3335::=
2704:30
2640:33
2634:27
2602:30
2587:30
2578:33
2572:27
2543:.
2531:13
2472:13
2412:16
2015:.
1981:.
1022:.
806:.
563:,
388:.
128:.
4369:.
4348:.
4318:.
4190:.
4185:)
4178:k
4174:b
4168:k
4162:1
4156:n
4152:a
4146:1
4140:n
4135:0
4132:=
4129:k
4119:(
4114:)
4111:b
4105:a
4102:(
4099:=
4094:n
4090:b
4081:n
4077:a
4064:R
4060:b
4056:a
4035:b
4031:a
4025:b
4021:a
3999:b
3993:a
3975:)
3970:b
3960:a
3955:(
3949:)
3944:b
3939:+
3934:a
3929:(
3926:=
3921:b
3911:b
3901:a
3891:a
3848:b
3844:a
3838:b
3832:a
3793:)
3790:N
3787:,
3784:b
3781:+
3778:a
3775:(
3752:)
3749:N
3746:,
3743:b
3737:a
3734:(
3711:N
3691:N
3671:b
3662:a
3642:N
3620:2
3616:b
3607:2
3603:a
3579:N
3559:)
3556:b
3548:i
3544:a
3540:(
3537:)
3534:b
3531:+
3526:i
3522:a
3518:(
3515:=
3510:2
3506:b
3497:2
3492:i
3488:a
3484:=
3481:N
3459:2
3455:b
3432:i
3428:x
3407:i
3404:+
3395:N
3381:i
3377:a
3356:N
3348:2
3343:i
3339:a
3330:i
3326:x
3291:2
3287:t
3280:s
3258:2
3254:t
3233:s
3213:,
3210:t
3190:a
3170:,
3167:0
3164:=
3161:u
3139:2
3135:t
3131:a
3125:2
3122:1
3116:+
3113:t
3110:u
3107:=
3104:s
3053:)
3050:k
3047:+
3044:n
3041:2
3038:(
3035:k
3030:=
3022:)
3019:n
3013:)
3010:k
3007:+
3004:n
3001:(
2998:(
2995:)
2992:n
2989:+
2986:)
2983:k
2980:+
2977:n
2974:(
2971:(
2966:=
2959:2
2955:n
2946:2
2942:)
2938:k
2935:+
2932:n
2929:(
2895:1
2892:+
2889:n
2886:2
2881:=
2873:)
2870:n
2864:)
2861:1
2858:+
2855:n
2852:(
2849:(
2846:)
2843:n
2840:+
2837:)
2834:1
2831:+
2828:n
2825:(
2822:(
2817:=
2810:2
2806:n
2797:2
2793:)
2789:1
2786:+
2783:n
2780:(
2763:n
2759:n
2741:.
2726:=
2721:2
2717:3
2708:2
2681:2
2677:b
2668:2
2664:a
2611:)
2608:3
2605:+
2599:(
2596:)
2593:3
2584:(
2581:=
2511:4
2508:+
2503:3
2478:.
2468:)
2465:4
2457:3
2452:(
2449:5
2439:=
2406:3
2401:)
2398:4
2390:3
2385:(
2382:5
2375:=
2346:2
2342:4
2333:2
2327:3
2319:)
2316:4
2308:3
2303:(
2300:5
2293:=
2266:)
2263:4
2255:3
2250:(
2247:)
2244:4
2241:+
2236:3
2231:(
2226:)
2223:4
2215:3
2210:(
2207:5
2200:=
2173:4
2165:3
2158:4
2150:3
2134:4
2131:+
2126:3
2120:5
2114:=
2087:4
2084:+
2079:3
2073:5
2043:4
2040:+
2035:3
2029:5
1969:)
1966:i
1963:2
1957:z
1954:(
1934:)
1931:i
1928:2
1925:+
1922:z
1919:(
1896:)
1893:i
1890:2
1884:z
1881:(
1878:)
1875:i
1872:2
1869:+
1866:z
1863:(
1860:=
1837:2
1833:)
1829:i
1826:2
1823:(
1815:2
1811:z
1807:=
1797:)
1785:1
1779:=
1774:2
1770:i
1747:2
1743:i
1739:4
1731:2
1727:z
1723:=
1702:4
1699:+
1694:2
1690:z
1666:4
1663:+
1658:2
1654:z
1610:b
1607:a
1604:4
1601:=
1598:)
1595:b
1592:2
1589:(
1586:)
1583:a
1580:2
1577:(
1574:=
1571:)
1568:b
1565:+
1562:a
1556:b
1553:+
1550:a
1547:(
1544:)
1541:b
1535:a
1532:+
1529:b
1526:+
1523:a
1520:(
1517:=
1512:2
1508:)
1504:b
1498:a
1495:(
1487:2
1483:)
1479:b
1476:+
1473:a
1470:(
1444:)
1441:1
1438:+
1435:y
1432:+
1429:x
1426:(
1423:)
1420:y
1414:x
1411:(
1408:=
1405:y
1399:x
1396:+
1393:)
1390:y
1384:x
1381:(
1378:)
1375:y
1372:+
1369:x
1366:(
1363:=
1360:y
1354:x
1351:+
1346:2
1342:y
1333:2
1329:x
1305:)
1302:y
1296:x
1293:(
1290:)
1287:y
1284:+
1281:x
1278:(
1258:y
1252:x
1249:+
1244:2
1240:y
1231:2
1227:x
1203:)
1200:1
1194:x
1191:(
1188:)
1185:1
1182:+
1179:x
1176:(
1173:)
1170:1
1167:+
1162:2
1158:x
1154:(
1151:=
1148:)
1145:1
1137:2
1133:x
1129:(
1126:)
1123:1
1120:+
1115:2
1111:x
1107:(
1104:=
1101:1
1093:4
1089:x
1065:1
1057:4
1053:x
1010:)
1007:b
1001:a
998:(
995:)
992:b
989:+
986:a
983:(
980:=
975:2
971:b
962:2
958:a
937:)
934:b
928:a
925:(
922:)
919:b
916:+
913:a
910:(
890:b
884:a
864:b
861:+
858:a
836:2
832:b
823:2
819:a
794:)
791:b
785:a
782:(
779:)
776:b
773:+
770:a
767:(
764:=
759:2
755:b
746:2
742:a
721:)
718:b
712:a
709:(
706:)
703:b
700:+
697:a
694:(
674:)
671:b
665:a
662:(
659:b
656:+
653:)
650:b
644:a
641:(
638:a
616:2
612:b
603:2
599:a
569:R
565:b
561:a
544:0
541:=
538:b
535:a
529:a
526:b
501:2
497:b
488:2
484:a
471:.
457:2
453:b
446:b
443:a
437:a
434:b
431:+
426:2
422:a
408:R
404:b
400:a
396:R
360:2
356:b
347:2
343:a
339:=
336:)
333:b
327:a
324:(
321:)
318:b
315:+
312:a
309:(
283:0
280:=
277:b
274:a
268:a
265:b
233:2
229:b
222:b
219:a
213:a
210:b
207:+
202:2
198:a
194:=
191:)
188:b
182:a
179:(
176:)
173:b
170:+
167:a
164:(
109:)
106:b
100:a
97:(
94:)
91:b
88:+
85:a
82:(
79:=
74:2
70:b
61:2
57:a
20:)
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