17e13f58-9cf7-4bfc-b062-ebdf523e017c 35 4 5 decimal

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Some exercise Diophantine ...

isolve ... eval SYMBOLIC solve Diophantine equations for integer solutions

x y f 3 x 2^* 4 y * - 7 - :

n sol x y f n isolve maple :

n sol y 1 - 6 n * 1 - 2 n * + (* + = x 14 n * - = y 56 n * 3 - 2 n * + (* + = x 34 n * - = 21 mat

n Sol n sol mat2sys :

n Sol y 1 - 6 n * 1 - 2 n * + (* + = x 14 n * - = y 56 n * 3 - 2 n * + (* + = x 34 n * - = 41 sys

n Sol 2 el x 14 n * - =

x 14 n * - (≡

n Sol 4 el x 34 n * - =

x 34 n * - (≡

n Sol 1 el y 1 - 6 n * 1 - 2 n * + (* + =

y 1 - 6 n * 1 - 2 n * + (* + (≡

n Sol 3 el y 56 n * 3 - 2 n * + (* + =

y 56 n * 3 - 2 n * + (* + (≡

n 3 - 3 range : i 1 n rows range SolX i el 14 n * - 34 n * - 21 mat vectorize : for SolX 31 line

n 3 - 3 range : j 1 n rows range SolY j el 1 - 6 n * 1 - 2 n * + (* + 56 n * 3 - 2 n * + (* + 21 mat vectorize : for SolY 31 line

UV U V augment :

U SolX 1 el : V SolY 1 el : 12 mat

X 14 n * - 34 n * - 21 mat vectorize :

Y 1 - 6 n * 1 - 2 n * + (* + 56 n * 3 - 2 n * + (* + 21 mat vectorize :

sanity X Y f U V f - :

Sanity Mathcad UV 131521 mat 12516721 mat 91121 mat 598921 mat 5721 mat 173521 mat 1321 mat 1 - 521 mat 3 - 1 - 21 mat 51 - 21 mat 7 - 5 - 21 mat 351721 mat 11 - 9 - 21 mat 895921 mat 72 mat

U V f 2391222321 mat 2655253521 mat 2823275121 mat 2895287121 mat 2871289521 mat 2751282321 mat 2535265521 mat 71 mat

sanity 0021 mat 0021 mat 0021 mat 0021 mat 0021 mat 0021 mat 0021 mat 71 mat

Solving the Diophantine equation

4 x * 5 y * + 7 z * + 3 t * + 16 ≡ isolve maple 312_N1 * - (* - 4_N2 * + 7_N3 * + (-_N1_N2 1_N2 + 2_N1 * + 3_N3 * + 41 mat

t x y z 41 mat 36 Z1 * - 4 Z2 * - 7 Z3 * - Z1 Z2 12 Z1 * + Z2 + 3 Z3 * + 41 mat ≡

Do NOT nest the construct

x y q 12 x * 42 y * + 6 - :

x y q z isolve maple 37 z * + (- 12 z * + 21 mat

z 3 - 3 range : i 1 z rows range SolX i el 37 z * + (- : for SolX 31 line

z 3 - 3 range : i 1 z rows range SolY i el 12 z * + : for SolY 31 line

SolX 1 el 181143 - 10 - 17 - 24 - 71 mat

SolY 1 el 5 - 3 - 1 - 135771 mat

SolX 1 el SolY 1 el q 000000071 mat

x y eq2 36 x * 56 y * + 8 - :

x y eq2 w isolve maple 237 w * + (* - 49 w * + 21 mat

w 3 - 3 range : i 1 w rows range SolX i el 237 w * + (* - : for SolX 31 line

w 3 - 3 range : i 1 w rows range SolY i el 49 w * + : for SolY 31 line

SolX 1 el 362286 - 20 - 34 - 48 - 71 mat

SolY 1 el 23 - 14 - 5 - 413223171 mat

SolX 1 el SolY 1 el eq2 000000071 mat

x y eq3 3 x * 4 y * - 7 - :

3 x * 4 y * - 7 - λ isolve maple 23 λ * + 54 λ * + 21 mat

λ 3 - 3 range : i 1 λ rows range SolX i el 54 λ * + : for SolX 31 line

λ 3 - 3 range : i 1 λ rows range SolY i el 23 λ * + : for SolY 31 line

SolX 1 el 7 - 3 - 159131771 mat

SolY 1 el 7 - 4 - 1 - 2581171 mat

SolX 1 el SolY 1 el eq3 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 71 mat

x y eq1 30 x * 56 y * + 8 - :

x y eq1 ω isolve maple 467 ω * + (* - 1315 ω * + 21 mat

ω 3 - 3 range : i 1 ω rows range SolX i el 467 ω * + (* - : for SolX 31 line

ω 3 - 3 range : i 1 ω rows range SolY i el 1315 ω * + : for SolY 31 line

SolX 1 el 6032424 - 52 - 80 - 108 - 71 mat

SolY 1 el 32 - 17 - 2 - 1328435871 mat

SolX 1 el SolY 1 el eq1 000000071 mat

Diophantine function x y f 3 x 2^* 4 y * - 7 - :

Smath solve & plot Diophantine

n sol x y f n isolve maple :

slide the tractor-graph in reverse direction n 3.5 :

Diophantine Segments XY 14 n * - 34 n * - 21 mat 1 - 6 n * - 12 n 2^* + 518 n * - 12 n 2^* + 21 mat augment :

Diophantine solutions wrt 'n' 14 n * - 34 n * - 21 mat 1 - 6 n * - 12 n 2^* + 518 n * - 12 n 2^* + 21 mat f 36350721 mat

XY 13 - 12511 - 8922 mat

Diophantine Generatrix x y q 3 x 2^* 4 y * - 7 - :

The Diophantine segment supports fractional 'n'. Interesting to animate ... q & f plot same

integer n fractional n 21 mat # # 3 - 2 - 1 - 0123 # # 111 mat typical: n 2.5 21 mat ≡

x Interp XY 1 col XY 2 col x cinterp :

x XY 11 el 1 + : 12 -

x Interp 107

xy x x Interp augment : 12 - 10712 mat

plot f XY xy 31 sys :

XY 13 - 12511 - 8922 mat

x y axes x y / - :

xy 12 - 10712 mat

plot axes 2 1 sys

f 2 a * 2 c * ≡ 4 a * c e + ≡ 6 a * 2 b * + 4 c * d + 4 e * + ≡ 2 b * 2 d * 2 e * + ≡ 41 mat :

seed a 1 ≡ b 1 ≡ c 1 ≡ d 1 ≡ 41 mat :

e 3 :

a b c d 41 mat f seed FindRoot (: 171441 mat

a b c d e 51 mat 1714351 mat

a b + c d + e + 21 mat 8821 mat

Exponential threshold 15
... show Trailing '0' sanity 2 a * 2 c * - 4 a * c e + (- 6 a * 2 b * + 4 c * d + 4 e * + (- 2 b * 2 d * 2 e * + (- 41 mat : 0.000000000000 0.000000000001 0.000000000001 - 0.000000000001 41 mat