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# Sources and conventions for T503
The quantity is one scalar: beta = lim (2n) E_(2n)(|x|; [-1,1]),
where E_m is the least supremum error over real polynomials of degree at
most m. It has no parameters. The normalisation is the full polynomial
degree, not half the degree. Symmetrising an approximant cannot increase its
error against the even target |x|. This proves E_(2n+1)=E_(2n), and the
substitution t=x^2 identifies E_(2n) with degree-n approximation of sqrt(t)
on [0,1]. Consequently lim m E_m gives the same constant.
Primary source: R. S. Varga and A. J. Carpenter, On the Bernstein Conjecture
in Approximation Theory, Constructive Approximation 1 (1985), 333–348,
DOI 10.1007/BF01890040. Author-hosted scan:
https://www.math.kent.edu/~varga/pub/paper_150.pdf
The scan was downloaded and pages 333–338, 345–348 were inspected as
images. Equation (1.18), p. 335, and equation (5.3), p. 346, print:
0.28016 94990 23869 13303 64364 91230 67200 00424 82139 81236
These are 50 estimated decimal places. Section 2 computes E_(2n), n=1..52,
with a Remez iteration using 100-digit arithmetic. Page 338 estimates at
least 95 accurate decimal places for those finite-degree errors. Section 5
extrapolates in 1/n^2. Of the 210 entries in the last 20 columns of the
Richardson table, 182 agree with all 50 displayed digits; this is numerical
evidence, not a proof of a 50-digit error bound for the limit. Equation
(1.16) separately states rigorous bounds approximately 0.2801685460 and
0.2801733791. An outward rounding gives 0.2801685 < beta < 0.2801734.
OEIS A073001 was retrieved by its b-file, which contains exactly 50 digits
at indices 0..49. It is a second transcription of the same calculation,
not an independent high-precision computation. The generator retains only
these 50 digits; it does not pad to 100 digits.
Finch, Mathematical Constants (Cambridge University Press, 2003), section
4.4, pp. 257–259, is the common family reference. Its bibliographic location
was confirmed against MathWorld and OEIS. The book itself was not available
for a direct numerical check. Mathematical claims and the digits rely on
the directly inspected Varga–Carpenter paper.
The independent checker is a new Remez implementation for even polynomials
in x, using arbitrary precision arithmetic, safeguarded Newton searches for
the residual zeros and extrema, and an independent Richardson tableau. It
first checks the exact minimax polynomial x^2+1/8 and error 1/8. For each n,
it requires alternating residuals at n+2 ordered points on [0,1], a small
spread of their magnitudes, and n positive roots of the residual derivative.
The derivative has n coefficient sign changes, so Descartes' rule rules out
additional positive critical points. The arithmetic checks are numerical,
not interval certificates. The extrapolation assumption remains heuristic.