Pell episode

The Pell number is a mathematical sequence of positive whole numbers , the Pell numbers (Engl. Pell numbers ), as well as the Pell numbers second type (Engl. Companion Pell numbers ). It takes its name from the English mathematician John Pell (1611–1685).

Pell sequence / numbers

The sequence is defined recursively by:

${\ displaystyle P (n) = {\ begin {cases} 0, & {\ text {if}} n = 0; \\ 1, & {\ text {if}} n = 1; \\ 2P (n- 1) + P (n-2) & {\ text {otherwise}} \ end {cases}}}$

That means in words:

• the values zero and one are specified for the first two numbers
• each additional number is calculated by doubling the direct predecessor and then adding the previous one.

The first numbers in the sequence are (if you start counting with ): ${\ displaystyle n}$${\ displaystyle n = 0}$

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, ... (sequence A000129 in OEIS )

The Pell sequence can also be interpreted as a special case of the general Lucas sequence with and : ${\ displaystyle U_ {n} (P, Q)}$${\ displaystyle P = 2}$${\ displaystyle Q = -1}$

${\ displaystyle f_ {n} = U_ {n} (2, -1)}$

Silver cut

The following applies to the limit value of the ratio of two consecutive numbers in the Pell sequence:

${\ displaystyle \ delta _ {S}: = \ lim _ {n \ to \ infty} {\ frac {P (n)} {P (n-1)}} = 1 + {\ sqrt {2}}}$

This number is called the silver ratio in analogy to the golden ratio of the Fibonacci sequence .

Derivation of the numerical value

The following limit value must be determined: ${\ displaystyle L: = \ lim _ {n \ to \ infty} {\ frac {P (n)} {P (n-1)}}}$

With follows: ${\ displaystyle P (n) = 2 \ cdot P (n-1) + P (n-2)}$

${\ displaystyle L = \ lim _ {n \ to \ infty} {\ frac {2 \ cdot P (n-1) + P (n-2)} {P (n-1)}} = \ lim _ { n \ to \ infty} {\ frac {2 \ cdot P (n-1)} {P (n-1)}} + \ lim _ {n \ to \ infty} {\ frac {P (n-2) } {P (n-1)}} = 2+ \ lim _ {n \ to \ infty} {\ frac {P (n-2)} {P (n-1)}}}$

With ${\ displaystyle L = \ lim _ {n \ to \ infty} {\ frac {P (n-1)} {P (n-2)}}}$

further follows . This gives the quadratic equation${\ displaystyle L = 2 + {\ tfrac {1} {L}}}$ ${\ displaystyle L ^ {2} -2L-1 = 0}$

with the two solutions     and   . ${\ displaystyle L_ {1} = 1 + {\ sqrt {2}}}$${\ displaystyle L_ {2} = 1 - {\ sqrt {2}}}$

Since of these two values ​​only the positive one comes into question for the limit value, it follows:

${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {P (n)} {P (n-1)}} = 1 + {\ sqrt {2}}}$

Closed form of the Pell series

In the section Deriving the numerical value , the ratio of two consecutive numbers of the Pell sequence was shown for the limit values:

${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {P (n)} {P (n-1)}} = 1 + {\ sqrt {2}}}$   and   .${\ displaystyle \ lim _ {n \ to \ infty} {\ frac {P (n)} {P (n-1)}} = 1 - {\ sqrt {2}}}$

Be and real constants. Then meet the geometric sequences${\ displaystyle c_ {1}}$${\ displaystyle c_ {2}}$

${\ displaystyle P_ {1} (0): = c_ {1} \ quad P_ {1} (n): = c_ {1} (1 + {\ sqrt {2}}) ^ {n} \ quad n \ in \ mathbb {N}}$   and
${\ displaystyle P_ {2} (0): = c_ {2} \ quad P_ {2} (n): = c_ {2} (1 - {\ sqrt {2}}) ^ {n} \ quad n \ in \ mathbb {N}}$

the recursion formulas

${\ displaystyle P_ {1} (n) = 2P_ {1} (n-1) + P_ {1} (n-2)}$   and
${\ displaystyle P_ {2} (n) = 2P_ {2} (n-1) + P_ {2} (n-2)}$.

Their linear combination also fulfills the Pell recursion. ${\ displaystyle P_ {l} (n): = c_ {1} (1 + {\ sqrt {2}}) ^ {n} + c_ {2} (1 - {\ sqrt {2}}) ^ {n }}$

The following initial values ​​must apply to the Pell sequence:    and    . ${\ displaystyle P (0) = 0}$${\ displaystyle P (1) = 1}$

Inserted in results in the following system of equations: ${\ displaystyle P_ {l} (n)}$

${\ displaystyle P_ {l} (0) = c_ {1} + c_ {2} = 0}$   and
${\ displaystyle P_ {l} (1) = c_ {1} (1 + {\ sqrt {2}}) + c_ {2} (1 - {\ sqrt {2}}) = 1}$

with the solutions    and   ${\ displaystyle c_ {1} = {\ frac {1} {2 {\ sqrt {2}}}}}$${\ displaystyle c_ {2} = - {\ frac {1} {2 {\ sqrt {2}}}}}$

This gives the closed form of the Pell sequence:

${\ displaystyle P (n) = {\ frac {(1 + {\ sqrt {2}}) ^ {n} - (1 - {\ sqrt {2}}) ^ {n}} {2 {\ sqrt { 2}}}}.}$

Generating function of the Pell sequence

The generating function of the Pell sequence is:

${\ displaystyle {\ mathcal {P}} (x) = \ sum _ {n = 0} ^ {\ infty} P (n) \ cdot x ^ {n} = {\ frac {x} {1-2x- x ^ {2}}}.}$

This power series has the radius of convergence .${\ displaystyle {\ sqrt {2}} - 1}$

Derivation of the function

The generating function of the Pell sequence has the radius of convergence . The following applies to : ${\ displaystyle {\ sqrt {2}} - 1}$

${\ displaystyle | x | <{\ sqrt {2}} - 1}$${\ displaystyle P (n + 2) -2 \ cdot P (n + 1) -P (n) = 0, \ P (0) = 0 {\ text {and}} P (1) = 1}$

{\ displaystyle {\ begin {alignedat} {5} {\ mathcal {P}} (x) & = P (0) && + P (1) \ cdot x && + P (2) \ cdot x ^ {2} && + P (3) \ times x ^ {3} && + P (4) \ times x ^ {4} + \ dotsb \\ {- 2x} \ times {\ mathcal {P}} (x) & = && - 2 \ times P (0) \ times x && - 2 \ times P (1) \ times x ^ {2} && - 2 \ times P (2) \ times x ^ {3} && - 2 \ times P (3) \ cdot x ^ {4} - \ dotsb \\ {- x ^ {2}} \ cdot {\ mathcal {P}} (x) & = &&&& - P (0) \ cdot x ^ {2} && - P (1) \ cdot x ^ {3} && - P (2) \ cdot x ^ {4} - \ dotsb \\\ hline (1-2x-x ^ {2}) \ cdot {\ mathcal {P}} (x) & = P (0) && + P (1) \ times x-2 \ times P (0) \ times x \\ & = x \ end {alignedat}}}

Pell primes

A Pell prime is a Pell number that is prime. The smallest Pell prime numbers are:

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449, 4760981394323203445293052612223893281, ... (sequence A086383 in OEIS )

For these Pell primes the index is of the following: ${\ displaystyle n}$${\ displaystyle P (n)}$

2, 3, 5, 11, 13, 29, 41, 53, 59, 89, 97, 101, 167, 181, 191, 523, 929, 1217, 1301, 1361, 2087, 2273, 2393, 8093, 13339, 14033, 23747, 28183, 34429, 36749, 90197,… (Follow A096650 in OEIS )
Example 1:
It is and . Thus is a prime number. In fact, the index appears in the above list in 4th position because it leads to the fourth smallest Pell prime number .${\ displaystyle P (10) = 2378}$${\ displaystyle P (9) = 985}$${\ displaystyle P (11) = 2 \ cdot P (10) + P (9) = 2 \ cdot 2378 + 985 = 5741 \ in \ mathbb {P}}$${\ displaystyle n = 11}$${\ displaystyle P_ {11} = 5741}$

The following properties apply to Pell primes:

• If a Pell is prime, then the index is also a prime (the converse is incorrect, meaning that not every prime index leads to a Pell prime).${\ displaystyle P (n)}$${\ displaystyle n}$

Pell Numbers 2nd Kind / Companion Pell Series

Pell numbers of the 2nd type are also called Pell-Lucas numbers .

The sequence is defined recursively by:

${\ displaystyle Q (n) = {\ begin {cases} 2, & {\ text {if}} n = 0; \\ 2, & {\ text {if}} n = 1; \\ 2Q (n- 1) + Q (n-2) & {\ text {otherwise}} \ end {cases}}}$

That means in words:

• the value two is given for the first two numbers
• each additional number is calculated by doubling the direct predecessor and then adding the previous one.

The first numbers in the sequence are 2, 2, 6, 14, 34, 82, 198, 478, 1154, ... (sequence A002203 in OEIS )

The Companion Pell sequence can also be interpreted as a special case of the general Lucas sequence with and : ${\ displaystyle V_ {n} (P, Q)}$${\ displaystyle P = 2}$${\ displaystyle Q = -1}$

${\ displaystyle Q (n) = V_ {n} (2, -1)}$