History of Hindu Mathematics: A Source Book v2.c17.s14: 16. Solution of Nx²+1=y
History of Hindu Mathematics: A Source Book v2.c17.s14
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16. Solution of Nx²+1=y
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**Square-nature.** The indeterminate quadratic equation $Nx²± c=y²,$ is called by the Hindus *Varga-prakṛti* or *Kṛti-prakṛti*, meaning the “Square-nature.”¹ Bhâskara II (1150) states that the absolute number should be *rûpa*,² which means “unity” as well as “absolute number” in general. Kamalâkara (1658) says :
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¹ *Varga* = *kṛti* = “square” and *prakṛti* = “nature,” “principle,” “origin,” etc. Colebrooke has rendered the term *varga-prakṛti* as “Affected Square.”
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² “Tatra rûpakṣepapadârthaṁ tâvat”—*BBi*, p. 33.
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"Hear first the nature of the *varga-prakṛti* : in it the square (of a certain number) multiplied by a multiplier and then increased or diminished by an interpolator becomes capable of yielding a square-root."∗1
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∗1 *SiTVi*, xiii. 208.
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It was recognised that the most fundamental equation of this class is $Nx² + 1 = y²,$ where N is a non-square integer.
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**Origin of the Name.** As regards the origin of the name *varga-prakṛti*, Kṛṣṇa (1580) says: "That in which the *varga* (square) is the *prakṛti* (nature) is called the *varga-prakṛti*; for the square of *yâvat*, etc., is the *prakṛti* (origin) of this (branch of) mathematics. Or, because this (branch of) mathematics has originated from the number which is the *prakṛti* of the square of *yâvat*, etc., so it is called the *varga-prakṛti*. In this case the number which is the multiplier of the square of *yâvat*, etc., is denoted by the term *prakṛti*. (In other words) it is the coefficient of the square of the unknown."∗2 This double interpretation has been evidently suggested by the use of the term *prakṛti* by Bhâskara II in two contexts. He has denoted by it sometimes the quantity N of the above equation as in "There the number which is (associated) with the square of the unknown is the *prakṛti*; "∗3 and at other times x², as in "Supposing the square of one of the two unknowns to be the *prakṛti*."∗4 Other Hindu algebraists have, however, consistently employed the term *prakṛti* to denote N only.¹ Brahmagupta (628) uses the term *guṇaka* (multiplier) for the same purpose.² This latter term, together with its variation *guṇa*, appears occasionally also in later works.³
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∗2 See his commentary on the *Bîjagaṇita* of Bhâskara II.
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∗3 "Tatra varṇavarge yo'ṅkaḥ sâ prakṛtiḥ" (*BBi*, p. 100). Compare also "Tatra yâvattâvadvarge yo'ṅkaḥ sâ prakṛtiḥ" (p. 107); "Iṣṭaṁ hrasvaṁ tasya vargaḥ prakṛtyâ kṣuṇṇo..." (p. 33).
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∗4 "Tatraikâṁ varṇakṛtiṁ prakṛtiṁ prakalpya..." (*BBi*, p. 106). Compare also "Sarûpake varṇakṛti tu yatra tatrecchaikâṁ prakṛtiṁ prakalpya..." (p. 105).
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¹ For instance, Pṛthûdakasvâmî (860) writes: "The multiplier (of the square of the unknown) is known as the *prakṛti*;" Śrîpati (1039): "*Kṛter-guṇako prakṛtirbhṛśoktaḥ*" (*SiŚe*, xiv. 32); Kamalâkara: "*Guṇo yo râśi-vargasya saiva prakṛtirucyate.*"
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³ For instance, Śrîpati employs the term *guṇaka* (*SiŚe*, xiv. 32); Bhâskara II and Nârâyaṇa use *guṇa* (*BBi*, p. 42; *NBi*, I, R. 84).
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We presume that the name *varga-prakṛti* originated from the following consideration: The principle (*prakṛti*) underlying the calculations in this branch of mathematics is to determine a number (or numbers) whose nature (*prakṛti*) is such that its (or their) square (or squares, *varga*) or the simple number (or numbers) after certain specified operations will yield another number (or numbers) of the nature of a square. So the name is, indeed, very significant. This interpretation seems to have been intended, at any rate, by the earlier writers who used the term in a wider sense.⁴ It is perhaps noteworthy that we do not find in the works of Brahmagupta the use of the word *prakṛti* either in the sense of N or of x².
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² *BrSpSi*, xviii. 64.
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⁴ For instance, Brahmagupta seems to have considered the scope of the subject wide enough to include such equations as $x+y=u², x-y=v², xy+1=w²,$ amongst others (cf. *BrSpSi*, xviii. 72).
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**Technical Terms.** Of the various technical terms which are ordinarily used by the Hindu algebraists in connection with the Square-nature we have already dealt with the most notable one, *prakṛti*, together with its synonyms. Others have been explained by Pṛthûdakasvâmî (860) thus:
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"Here are stated for ordinary use the terms which
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are well known to people. The number whose square, multiplied by an optional multiplier and then increased or decreased by another optional number, becomes capable of yielding a square-root, is designated by the term the lesser root (*kaniṣṭha-pada*) or the first root (*âdya-mûla*). The root which results, after those operations have been performed, is called by the name the greater root (*jyeṣṭha-pada*) or the second root (*anya-mûla*). If there be a number multiplying both these roots, it is called the augmenter (*udvartaka*); and, on the contrary, if there be a number dividing the roots, it is called the abridger (*apavartaka*).”¹
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¹ See Pṛthûdakasvâmî’s commentary on *BrSpSi*, xviii. 64. In the equation Nx² ± c = y², x = lesser root, y = greater root, N = multiplier, and c = interpolator.
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“An optionally chosen number is taken as the lesser root (*hrasva-mûla*). That number, positive or negative, which being added to or subtracted from its square multiplied by the *prakṛti* (multiplier) gives a result yielding a square-root, is called the interpolator (*kṣepaka*). And this (resulting) root is called the greater root (*jyeṣṭha-mûla*).”²
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Similar passages occur in the works of Nârâyaṇa,³ Jñânarâja and Kamalâkara.⁴
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³ *NBi*, I, R. 72.
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⁴ *SiTVi*, xiii. 209.
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The terms ‘lesser root’ and ‘greater root’ do not appear to be accurate and happy. For if x = m, y = n be a solution of the equation Nx² + c = y², m will be less than n, if N and c are both positive. But if they are of opposite signs, the reverse will sometimes happen.⁵
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² *BBi*, p. 33.
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⁵ For instance, take the following example from Bhâskara II (*BBi*, p. 43) :
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Therefore, in the latter case, where m > n, it will be obviously ambiguous to call m the lesser root and n the greater root, as was the practice in later Hindu algebra. This defect in the prevalent terminology was noticed by Kṛṣṇa (1580). He explains it thus: "These terms are significant. Where the greater root is sometimes smaller than the lesser root owing to the interpolator being negative, there also it becomes greater than the lesser root after the application of the Principle of Composition."¹ The earlier terms, 'the first root' (*âdya-mûla*) for the value of x and 'the second root' or 'the last root' (*antya-mûla*) for the value of y, are quite free from ambiguity. Their use is found in the algebra of Brahmagupta (628).² The later terms appear in the works of his commentator Pṛthûdakasvâmî (860).
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¹ For example, by composition of the solution (1, 0) of the equation 13x²-13=y² with the solution (3/2, 11/2) of the equation 13x²+1=y², we obtain, after Bhâskara II, a new solution (11/2, 39/2) of the former, in which the greater root is greater than the lesser root. Similarly, by composition of the solution (2, 1) of the equation -5x²+21=y² with the solution (½, 3/2) of the equation -5x²+1=y², we get a new solution (1, 4) of the former satisfying the same condition.
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The interpolator is called by Brahmagupta *kṣepa*, *prakṣepa* or *prakṣepaka*.³ Śrîpati occasionally employs the synonym *kṣipti*.⁴ When negative, the interpolator is sometimes distinguished as 'the subtractive' (*śodhaka*).
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³ *BrSpSi*, xviii. 65.
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⁴ *SiŚe*, xiv. 32.
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$13x²-13=y².$ One solution of it is given by the author as x=1, y=0; so that here the lesser root is greater than the greater root. The same is the case in the solution x=2, y=1 of his example (*BBi*, p. 43) $-5x²+21=y².$ Brahmagupta gives the example (*BrSpSi*, xviii. 77) $3x²-800=y²,$ which has a solution (x=20, y=20) where the two roots are equal.
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² *BrSpSi*, xviii. 64, 66f.
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The positive interpolator is then called 'the additive.'¹
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¹ *BrSpSi*, xviii. 64-5.
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**Brahmagupta's Lemmas.** Before proceeding to the general solution of the Square-nature Brahmagupta has established two important lemmas. He says :
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"Of the square of an optional number multiplied by the *guṇaka* and increased or decreased by another optional number, (extract) the square-root. (Proceed) twice. The product of the first roots multiplied by the *guṇaka* together with the product of the second roots will give a (fresh) second root ; the sum of their cross-products will be a (fresh) first root. The (corresponding) interpolator will be equal to the product of the (previous) interpolators."²
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The rule is somewhat cryptic because the word *dvidhā* (twice) has been employed with double implication. According to one, the earlier operations of finding roots are made on two optional numbers with two optional interpolators, and with the results thus obtained the subsequent operations of their composition are performed. According to the other implication of the word, the earlier operations are made with one optionally chosen number and one interpolator, and the subsequent ones are carried out after the repeated statement of those roots for the second time. It is also implied that in the composition of the quadratic roots their products may be added together or subtracted from each other.
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That is to say, if x = α, y = β be a solution of the equation Nx² + k = y², and x = α', y = β' be a solution of Nx² + k' = y², then, according to the above, x=αβ'±α'β, y=ββ'± Nαα' is a solution of the equation Nx²+kk'=y². In other words, if N\alpha²+k &= \beta², / N\alpha'²+k' &= \beta'², then N(αβ'±α'β)²+kk'=(ββ'± N.α')². I
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² *BrSpSi*, xviii. 64-5.
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In particular, taking α=α', β=β', and k=k', Brahmagupta finds from a solution x=α, y=β of the equation $Nx²+k=y²,$ a solution x=2αβ, y=β²+Nα² of the equation $Nx²+k²=y².$ That is, if $Nα²+k=β²,$ then $N(2αβ)²+k²=(β²+Nα²)². II$
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This result will be hereafter called *Brahmagupta's Corollary*.
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**Description by Later Writers.** Brahmagupta's Lemmas have been described by Bhâskara II (1150) thus :
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“Set down successively the lesser root, greater root and interpolator ; and below them should be set down in order the same or another (set of similar quantities). From them by the Principle of Composition can be obtained numerous roots. Therefore, the Principle of Composition will be explained here. (Find) the two cross-products of the two lesser and the two greater roots ; their sum is a lesser root. Add the product of the two lesser roots multiplied by the *prakṛti* to the product of the two greater roots ; the sum will be a greater root. In that (equation) the interpolator will be
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the product of the two previous interpolators. Again the difference of the two cross-products is a lesser root. Subtract the product of the two lesser roots multiplied by the *prakṛti* from the product of the two greater roots; (the difference) will be a greater root. Here also, the interpolator is the product of the two (previous) interpolators.”¹
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¹ *BBi*, p. 34.
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Statements similar to the above are found in the works of Nârâyaṇa² (1350), Jñanarâja (1503) and Kamalâkara³ (1658).
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² *NBi*, I, R. 72-75½.
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³ *SiTVi*, xiii. 210-214.
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**Principle of Composition.** The above results are called by the technical name, *Bhâvanâ* (demonstration or proof, meaning anything demonstrated or proved, hence theorem, lemma; the word also means composition or combination). They are further distinguished as *Samâsa Bhâvanâ* (Addition Lemma or Additive Composition) and *Antara Bhâvanâ* (Subtraction Lemma or Subtractive Composition). Again, when the *Bhâvanâ* is made with two equal sets of roots and interpolators, it is called *Tulya Bhâvanâ* (Composition of Equals) and when with two unequal sets of values, *Atulya Bhâvanâ* (Composition of Unequals). Kṛṣṇa has observed that when it is desired to derive roots of a Square-nature, larger in value, one should have recourse to the Addition Lemma and for smaller roots one should use the Subtraction Lemma.
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Brahmagupta's Lemmas were rediscovered and recognised as important by Euler in 1764 and by Lagrange in 1768.
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**Proof.** The proof of Brahmagupta's Lemmas has been given by Kṛṣṇa substantially as follows :
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We have $Nα²+k=β²,$ $Nα'²+k'=β'².$ Multiplying the first equation by β'², we get $Nα²β'²+kβ'²=β²β'².$
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Now, substituting the value of the factor β'² of the interpolator from the second equation, we get $Nα²β'²+k(Nα'²+k')=β²β'²,$ or $Nα²β'²+Nkα'²+kk'=β²β'².$ Again, substituting the value of k from the first equation in the second term of the left-hand side expression, we have $Nα²β'²+Nα'²(β²-Nα²)+kk'=β²β'²,$ or $N(α²β'²+α'²β²)+kk'=β²β'²+N²α²α'².$ Adding ± 2Nαβα'β' to both sides, we get $N(αβ'± α'β)²+kk'=(ββ'± Nαα')².$
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Brahmagupta's Corollary follows at once from the above by putting α'=α, β'=β and k'=k.
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**General Solution of the Square-Nature.** It is clear from Brahmagupta's Lemma (I) that when two solutions of the Square-nature, $Nx²+1=y²,$ are known, any number of other solutions can be found. For, if the two solutions be (a, b) and (a', b'), then two other solutions will be $x=ab'± a'b, y=bb'± Naa'.$ Again, composing this solution with the previous ones, we shall get other solutions. Further, it follows from Brahmagupta's Corollary that if (a, b) be a solution of the equation, another solution of it is (2ab, b²+Na²). Hence, in order to obtain a set of solutions of the Square-nature it is necessary to obtain only one solution of it. For, after having obtained that, an infinite number of other solutions can be found by the repeated application of the Principle of Composition. Thus Śrîpati (1039) observes : “There will be an infinite (set of two roots).”¹ Bhâskara II (1150) remarks : “Here (*i.e.*, in the solution of the Square-nature) the roots are infinite by virtue of (the infinitely repeated application of) the Principle of Composition as well as of (the infinite variety of) the optional values (of the first roots).”² Nârâyaṇa (1350) writes, “By the Principle of Composition of equal as well as unequal sets of roots, (will be obtained) an infinite number of roots.”³
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¹ *SiŚe*, xiv. 33.
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³ *NBi*, I, R. 78. Compare also *SiTVi*, xiii. 217.
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Modern historians of mathematics are incorrect in stating that Fermat (1657) was the first to assert that the equation Nx² + 1 = y², where N is a non-square integer, has an unlimited number of solutions in integers.⁴ The existence of an infinite number of integral solutions was clearly mentioned by Hindu algebraists long before Fermat.
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⁴ Smith, *History*, II, p. 453.
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**Another Lemma.** Brahmagupta says :
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“On dividing the two roots (of a Square-nature) by the square-root of its additive or subtractive, the roots for the interpolator unity (will be found).”⁵
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⁵ *BrSpSi*, xviii. 65.
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That is to say, if x = α, y = β be a solution of the equation $Nx² + k² = y²,$ then x = α/k, y = β/k is a solution of the equation $Nx² + 1 = y².$
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² “Ihânantyaṁ bhâvanâbhistatheṣṭataḥ”—*BBi*, p. 34.
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This rule has been restated in a different way thus :
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"If the interpolator is that divided by a square then the roots will be those multiplied by its square-root."¹
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¹ *BrSpSi*, xviii. 70.
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That is, suppose the Square-nature to be $Nx² ± p²d = y²,$ so that its interpolator p²d is exactly divisible by the square p². Then, putting therein u=x/p, v=y/p, we derive the equation $Nu² ± d = v²,$ whose interpolator is equal to that of the original Square-nature divided by p². It is clear that the roots of the original equation are p times those of the derived equation.
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Bhâskara II writes:
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"If the interpolator (of a Square-nature) divided by the square of an optional number be the interpolator (of another Square-nature), then the two roots (of the former) divided by that optional number will be the roots (of the other). Or, if the interpolator be multiplied, the roots should be multiplied."²
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The same rule has been stated in slightly different words by Nârâyaṇa³ and Kamalâkara.⁴ Jñânarâja simply observes:
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³ *NBi*, I, R. 76-76½.
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⁴ *SiTVi*, xiii. 215.
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"If the interpolator (of a Square-nature) be divided by the square of an optional number then its roots will be divided by that optional number."
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Thus we have, in general, if x=α, y=β be a solution of the equation $Nx² ± k = y²,$ x=α/m, y=β/m is a solution of the equation $Nx²± k/m²=y²;$ and x=na, y=nβ is a solution of the equation $Nx²± n²k=y²,$ where m, n are arbitrary rational numbers.
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² *BBi*, p. 34.
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By this Lemma, the solutions of the Square-natures (*i*) 6x²+12=y², (*ii*) 6x²+75=y², and (*iii*) 6x²+300=y², can be derived, as shown by Bhâskara II,¹ from those of $6x²+3=y²,$ since 12=2².3, 75=5².3, and 300=10².3. How to solve this latter equation will be indicated later on.
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¹ *BBi*, p. 41.
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**Rational Solution.** In order to obtain a first solution of Nx²+1=y² the Hindus generally suggest the following tentative method: Take an arbitrary small rational number α, such that its square multiplied by the *guṇaka* N and increased or diminished by a suitably chosen rational number k will be an exact square. In other words, we shall have to obtain empirically a relation of the form $Na²± k=β²,$ where α, k, β are rational numbers. This relation will be hereafter referred to as the *Auxiliary Equation*. Then by Brahmagupta's Corollary, we get from it the relation $N(2αβ)²+k²=(β²+Nα²)²,$ or $N((2αβ)/k)²+1=((β²+Nα²)/k)².$
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Hence, one rational solution of the equation Nx²+1=y² is given by $x = (2αβ)/k, y = (β²+Nα²)/k.(A)$
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**Śrîpati's Rational Solution.** Śrîpati (1039) has shown how a rational solution of the Square-nature can be obtained more easily and directly without the intervention of an auxiliary equation. He says:
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"Unity is the lesser root. Its square multiplied by the *prakṛti* is increased or decreased by the *prakṛti* combined with an (optional) number whose square-root will be the greater root. From them will be obtained two roots by the Principle of Composition."¹
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¹ *SiŚe*, xiv. 33.
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If m² be a rational number optionally chosen, we have the identity $N.1²+(m²-N)=m²,$ or $N.1²-(N-m²)=m².$ Then, applying Brahmagupta's Corollary to either, we get $N(2m)²+(m²~ N)²=(m²+N)²;$ $∴ N(2m/(m²~ N))²+1=((m²+N)/(m²~ N))².$ Hence $x = 2m/(m²~ N), y = (m²+N)/(m²~ N),(B)$ where m is any rational number, is a solution of the equation $Nx²+1=y².$
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The above solution reappears in the works of later Hindu algebraists. Bhâskara II says :
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"Or divide twice an optional number by the difference between the square of that optional number and the *prakṛti*. This (quotient) will be the lesser root (of a Square-nature) when unity is the additive. From that (follows) the greater root."\∗1
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\∗1 *BBi*, p. 34.
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Nârâyaṇa states :
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"Twice an optional number divided by the difference between the square of that optional number and the *guṇaka* will be the lesser root. From that with the additive unity determine the greater root."\∗2
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\∗2 *NBi*, I, R. 77f.
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Similar statements are found also in the works of Jñânarâja and Kamalâkara.\∗3
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\∗3 *SiTVi*, xiii. 216.
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If m be an optional number, it is stated that 2m/(m² ~ N) is a lesser root of Nx² + 1 = y². Then, substituting that value of x in the equation, we get $y² = N(2m/(m² ~ N))² + 1,$ $= ((m² + N)/(m² ~ N))².$
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Hence the greater root is $y = (m² + N)/(m² ~ N).$
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The same solution will be obtained by assuming $y = mx - 1.$
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Kṛṣṇa points out that it can also be found thus : $4Nm² = (m² + N)² - (m² ~ N)², identically.$ $∴ 4Nm² + (m² ~ N)² = (m² + N)²,$ or N(2m/(m²~ N))²+1=((m²+N)/(m²~ N))².
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His remark that this method does not require the help of the Principle of Composition shows that Bhâskara II and others obtained the solution in the way indicated by Srîpati.
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The above rational solution of the Square-nature has been hitherto attributed by modern historians of mathematics to Bhâskara II. But it is now found to be due to an anterior writer, Srîpati (1039). It was rediscovered in Europe by Brouncker (1657).
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**Illustrative Examples.** In illustration of the foregoing rules we give the following examples with their solutions from Bhâskara II.
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*Examples.* "Tell me, O mathematician, what is that square which multiplied by 8 becomes, together with unity, a square; and what square multiplied by 11 and increased by unity, becomes a square."¹
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¹ *BBi*, p. 35.
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That is to say, we have to solve
(1) 8x²+1=y²,
(2) 11x²+1=y².
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*Solutions.* "In the second example assume 1 as the lesser root. Multiplying its square by the *prakṛti*, namely 11, subtracting 2 and then extracting the square-root, we get the greater root as 3. Hence the statement for composition is² m=11 & l=1 & g=3 & i=-2 / & l=1 & g=3 & i=-2
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² The abbreviations are: m=multiplier, l=lesser root, g=greater root and i=interpolator. In the original they are respectively *pra*, *ka*, *jye*, and *kṣe*, the initial syllables of the corresponding Sanskrit terms.
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Proceeding as before we obtain the roots for the additive 4 : l = 6, g = 20, (for) i = 4. Then by the rule, 'If the interpolator (of a Square-nature) divided by the square of an optional number etc.,'¹ are found the roots for the additive unity: l = 3, g = 10 (for) i = 1. Whence by the Principle of Composition of Equals, we get the lesser and greater roots : l = 60, g = 199 (for) i = 1. In this way an infinite number of roots can be deduced.
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¹ *Vide supra*, p. 151.
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"Or, assuming 1 for the lesser root, we get for the additive 5 : l = 1, g = 4, (for) i = 5. Whence by the Principle of Composition of Equals, the roots are l = 8, g = 27, (for) i = 25. Then by the rule, 'If the interpolator (of a Square-nature) divided by the square of an optional number etc.,' taking 5 as the optional number, we get the roots for the additive unity : l = 8/5, g = 27/5, (for) i = 1. The statement of these for composition with the previous roots is {llll} m = II & l = 8/5 & g = 27/5 & i = 1 / & l = 3 & g = 10 & i = 1
v2.c17.s14.109
By the Principle of Composition the roots are obtained as: l = 161/5, g = 534/5 (for) i = 1.
v2.c17.s14.110
"Or composing according to the rule, 'The difference of the two cross-products is a lesser root etc.,' we get the roots : l = 1/5, g = 6/5 (for) i = 1. And so on in many ways.
v2.c17.s14.111
"The two roots for the additive unity will now be found in a different way by the rule, 'Or divide twice an optional number by the difference between the square of that optional number and the *prakṛti* etc.' Here, in the first example, assume the optional number to be 3. Its square is 9; multiplier is 8; their difference is 1;
v2.c17.s14.112
dividing by this twice the optional number, namely 6, we get the lesser root for the additive unity as 6. Whence, proceeding as before, the greater root comes out as 17.
v2.c17.s14.113
"In the same way, in the second example also, assuming the optional number to be 3, the lesser and greater roots are found to be (3, 10).
v2.c17.s14.114
"Thus, by virtue of (the infinite variety of) the optional values as well as of (the infinitely repeated application of) the Principle of Additive and Subtractive Compositions, an infinite number of roots (may be found)."¹
v2.c17.s14.115
¹ The original is, "Evamiṣṭavaśât samâsântarabhâvanâbhyâṁ ca padânâmânantyam." (*BBi*, p. 36).
v2.c17.s14.116
**Solution in Positive Integers.** As has been stated before, the aim of the Hindus was to obtain solutions of the Square-nature in positive integers; so its first solution must be integral. But neither the tentative method of Brahmagupta nor that of Śrîpati is of much help in this direction, for they do not *always* yield the desired result. These authors, however, discovered that if the interpolator of the auxiliary equation in the tentative method be ± 1, ± 2 or ± 4, an integral solution of the equation Nx²+1=y² can always be found. Thus Śrîpati (1039) expressly observes, "If 1, 2 or 4 be the additive or subtractive (of the auxiliary equation) the lesser and greater roots will be integral (*abhinna*).²
v2.c17.s14.117
(*i*) If k=± 1; then the auxiliary equation will be³ $Na²± 1=β²,$ where α, β are integers. Then by Brahmagupta's Corollary, we get x = 2αβ, y = β² + Na² as the required first solution in positive integers of the equation Nx² + 1 = y².
v2.c17.s14.118
² "Dvyekâmbudhikṣepaviśodhanâbhyâṁ Syâtâmabhinne laghuvṛddhamûle." — *SiŚe*, xiv. 32. The Sanskrit word *abhinna* literally means "non-fractional."
v2.c17.s14.119
³ The special treatment of the equation Nx²-1=y² is given later on.
v2.c17.s14.120
(*ii*) Let k = ± 2; then the auxiliary equation is Na² ± 2 = β².
v2.c17.s14.121
By Brahmagupta's Corollary, we have N(2αβ)² + 4 = (β² + Na²)²,
v2.c17.s14.122
or N(αβ)² + 1 = ((β² + Na²)/2)².
v2.c17.s14.123
Hence the required first solution is x = αβ, y = ½(β² + Na²).
v2.c17.s14.124
Since Na² = β² 2, we have ½(β² + Na²) = β² 1 = a whole number.
v2.c17.s14.125
(*iii*) Now suppose k = + 4; so that Na² + 4 = β².
v2.c17.s14.126
With an auxiliary equation like this the first integral solution of the equation Nx² + 1 = y² is x = ½αβ, y = ½(β² - 2); if α is even; or x = ½α(β² - 1), y = ½β(β² - 3); if β is odd.
v2.c17.s14.127
Thus Brahmagupta says:
v2.c17.s14.128
“In the case of 4 as additive the square of the second root diminished by 3, then halved and multiplied by the second root will be the (required) second root; The square of the second root diminished by unity and
v2.c17.s14.129
then divided by 2 and multiplied by the first root will be the (required) first root (for the additive unity).”¹
v2.c17.s14.130
¹ *BrSpŚi*, xviii. 67.
v2.c17.s14.131
$Since Nα²+4=β², (1)$ $we have N((α)/2)²+1=((β)/2)². (2)$
v2.c17.s14.132
Then, by Brahmagupta's Corollary, we get $N((αβ)/2)²+1=((β²)/4+N(α²)/4)².$
v2.c17.s14.133
Substituting the value of N in the right-hand side expression from (1), we have $N((αβ)/2)²+1=((β²-2)/2)². (3)$
v2.c17.s14.134
Composing (2) and (3), $N(α)/2(β²-1)²+1=(β)/2(β²-3)².$
v2.c17.s14.135
Hence x=½αβ, y=½(β²-2); and x=½α(β²-1), y=½β(β²-3); are solutions of $Nx²+1=y².$
v2.c17.s14.136
If β be even, the first values of (x, y) are integral. If β be odd, the second values are integral.
v2.c17.s14.137
(*iv*) Finally, suppose k=-4; the auxiliary equation is Nα²-4=β². Then the required first solution in positive integers of Nx²+1=y² is x&=1/2\alpha\beta(\beta²+3)(\beta²+1), / y&=(\beta²+2){1/2(\beta²+3)(\beta²+1)-1}.
v2.c17.s14.138
Brahmagupta says :
v2.c17.s14.139
“In the case of 4 as subtractive, the square of the second is increased by three and by unity ; half the product of these sums and that as diminished by unity (are obtained). The latter multiplied by the first sum less unity is the (required) second root; the former multiplied by the product of the (old) roots will be the first root corresponding to the (new) second root.”
v2.c17.s14.140
The *rationale* of this solution is as follows : $Na² - 4 = β². 1$ $∴ N((α)/2)² - 1 = ((β)/2)².$
v2.c17.s14.141
Hence by Brahmagupta’s Corollary, we get N((\alpha\beta)/2)² + 1 &= ((\beta²)/4 + N(\alpha²)/4)² / &= {1/2(\beta² + 2)}². \tag
v2.c17.s14.142
Again, applying the Corollary, we have $N½αβ(β² + 2)² + 1 = ½(β⁴ + 4β² + 2)². 3$
v2.c17.s14.143
Now, by the Lemma, we obtain from (2) and (3) N{1/2\alpha\beta(\beta² &+ 3)(\beta² + 1)}² + 1 / &= [(\beta² + 2){1/2(\beta² + 3)(\beta² + 1) - 1}]².
v2.c17.s14.144
Hence x &= 1/2\alpha\beta(\beta² + 3)(\beta² + 1), / y &= (\beta² + 2){1/2(\beta² + 3)(\beta² + 1) - 1}, is a solution of Nx² + 1 = y².
v2.c17.s14.145
It can be proved easily that these values of x, y are integral. For, if β is even, β² + 2 is also even. Therefore, the above values of x, y are integral. If on the contrary β is odd, β² is also odd ; then β² + 1 and β² + 3 are even. Hence in this case also the above values are integral.
v2.c17.s14.146
¹ *BrSpSi*, xviii. 68.
v2.c17.s14.147
Putting p = αβ, q = β² + 2 we can write the above solution in the form $x = ½p(q² - 1),$ $y = ½q(q² - 3),$ in which it was found by Euler.
Linked texts (5)
- Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
- Bījagaṇita — Bhāskara II, Sanskrit
- Siddhāntatattvaviveka — Kamalākarabhaṭṭa 1633, Sanskrit
- Siddhāntaśekhara — Śrīpati, Sanskrit
- Siddhāntaśiromaṇi — Bhāskara II, Sanskrit